![Abstract
In this paper, we study all the elliptic integrable systems, in the sense of C.
L. Terng [66], that is to say, the family of all the m-th elliptic integrable systems
associated to a k
-symmetric space N = G/G0. Here m ∈ N and k
∈ N∗
are
integers. For example, it is known that the first elliptic integrable system asso-
ciated to a symmetric space (resp. to a Lie group) is the equation for harmonic
maps into this symmetric space (resp. this Lie group). Indeed it is well known
that this harmonic maps equation can be written as a zero curvature equation:
dαλ + 1
2 [αλ ∧ αλ] = 0, ∀λ ∈ C∗
, where αλ = λ−1
α
1 + α0 + λα
1 is a 1-form
on a Riemann surface L taking values in the Lie algebra g. This 1-form αλ is
obtained as follows. Let f : L → N = G/G0 be a map from the Riemann sur-
face L into the symmetric space G/G0. Then let F : L → G be a lift of f, and
consider α = F−1
.dF its Maurer-Cartan form. Then decompose α according to
the symmetric decomposition g = g0 ⊕ g1 of g : α = α0 + α1. Finally, we define
αλ := λ−1
α
1+α0+λα
1 , ∀λ ∈ C∗
, where α
1, α
1 are the resp. (1, 0) and (0, 1) parts of
α1. Then the zero curvature equation for this αλ, for all λ ∈ C∗
, is equivalent to the
harmonic maps equation for f : L → N = G/G0, and is by definition the first ellip-
tic integrable system associated to the symmetric space G/G0. Thus the methods
of integrable system theory apply to give generalised Weierstrass representations,
algebro-geometric solutions, spectral deformations, and so on. In particular, we
can apply the DPW method [23] to obtain a generalised Weierstrass representa-
tion. More precisely, we have a Maurer-Cartan equation in some loop Lie algebra
Λgτ = {ξ : S1
→ g|ξ(−λ) = τ(ξ(λ))}. Then we can integrate it in the corresponding
loop group and finally apply some factorization theorems in loop groups to obtain a
generalised Weierstrass representation: this is the DPW method. Moreover, these
methods of integrable system theory hold for all the systems written in the forms of
a zero curvature equation for some αλ = λ−m
α̂−m +· · ·+α̂0 +· · ·+λm
α̂m. Namely,
these methods apply to construct the solutions of all the m-th elliptic integrable
systems. So it is natural to ask what is the geometric interpretation of these sys-
tems. Do they correspond to some generalisations of harmonic maps? This is the
problem that we solve in this paper: to describe the geometry behind this family
Received by the editor April 8, 2010 and, in revised form, March 31, 2011.
Article electronically published on February 22, 2012; S 0065-9266(2012)00651-4.
2010 Mathematics Subject Classification. Primary 53B20, 53B35; Secondary 53B50.
Research supported successively by the DFG-Schwerpunkt Globale Differentialgeometrie,
grant DO 776 (in T.U. Munich), and then by the SFB - TR 71 (in Universität Tübingen).
Affiliation at time of publication: Institut Élie Cartan de Nancy, Université Henri Poincaré
Nancy 1, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cdex, France; email: khemar@iecn.u-nancy.fr.
c
2012 American Mathematical Society
ix](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-14-320.jpg)
![Introduction
In this paper, we give a geometric interpretation of all the m-th elliptic in-
tegrable systems associated to a k
-symmetric space N = G/G0 (in the sense of
C.L. Terng [66]).
Let g be a real Lie algebra and τ : g → g be an automorphism of finite order
k
. This automorphism admits an eigenspace decomposition gC
= ⊕j∈Zk gC
j , where
gC
j is the eigenspace of τ with respect to the eigenvalue ωj
k . We denote by ωk a
k
-th primitive root of unity. Moreover the automorphism τ defines a k
-symmetric
space N = G/G0. Furthermore let L be a Riemann surface.
The m-th elliptic integrable system associated to (g, τ) can be written as a zero
curvature equation
dαλ +
1
2
[αλ ∧ αλ] = 0, ∀λ ∈ C∗
,
where αλ =
m
j=0 λ−j
uj +λj
ūj =
m
j=−m λj
α̂j is a 1-form on the Riemann surface
L taking values in the Lie algebra g. The ”coefficient” uj is a (1, 0)-type 1-form on
L with values in the eigenspace gC
−j.
Moreover, we call the integer m the order of the system. Let us make precise
that the order m has nothing to do with the order of any PDE, but this is only the
maximal power on λ in the (finite) Fourier decomposition of αλ w.r.t. λ.
First, we remark that any solution of the system of order m is a solution of the
system of order m
, if m ≤ m
(and the automorphism τ is fixed). In other words,
the system of order m is a reduction of the system of order m
, if m ≤ m
.
Moreover, it turns out that we have to introduce the integer mk defined by
mk =
k
+ 1
2
=
k if k
= 2k
k + 1 if k
= 2k + 1
if k
1, and m1 = 0.
Then the general problem splits into three cases : the primitive case (m mk ),
the determined case (mk ≤ m ≤ k
− 1) and the underdetermined case (m ≥ k
).
0.1. The primitive systems.
The primitive systems have an interpretation in terms of F-holomorphic maps,
with respect to an f-struture F on the target space N = G/G0 (i.e. an endomor-
phism F satisfying F3
+ F = 0). More precisely:
• In the even case (k
= 2k), we have a fibration π: G/G0 → G/H over a
k-symmetric space M = G/H (defined by the square τ2
of the automorphism τ of
order k
defining N = G/G0). We also have a G-invariant splitting TN = H ⊕ V
corresponding to this fibration1
, where V = ker dπ. Moreover N is naturally
endowed with an f-structure F which defines a complex structure on the hori-
zontal subbundle H and vanishes on the vertical subbundle V. Furthermore, the
1i.e. H is a connection on this fibration.
1](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-16-320.jpg)
![2 IDRISSE KHEMAR
eigenspace decomposition of the order k
automorphism τ gives us some G-invariant
decomposition H = ⊕k−1
j=1 [mj], where mj ⊂ g is defined by mC
j = gC
−j ⊕ gC
j , and
[mj] ⊂ TN is the corresponding G-invariant subbundle. This allows to define, by
multiplying F on the left by the projections on the subbundles Hm
= ⊕m
j=1[mj],
a family of f-structures F[m]
, 1 ≤ m ≤ k − 1. Then the primitive system of
order m (m mk = k) associated to G/G0 is exactly the equation for F[m]
-
holomorphic maps. Therefore the solutions of the primitive systems are exactly the
F-holomorphic maps.
• In the odd case (k
= 2k+1), N = G/G0 is naturally endowed with an almost
complex structure J. Then the solutions of the primitive systems are exactly the J-
holomorphic curves. Moreover, in the same way as for the even case, the eigenspace
decomposition of τ provides a G-invariant decomposition TN = ⊕k
j=1[mj], which
allows to define a family of f-structures F[m]
, 1 ≤ m ≤ k, with F[k]
= J. Then
the primitive system of order m (m mk = k + 1) associated to G/G0 is exactly
the equation for F[m]
-holomorphic maps. In other words, the solutions of the
primitive system of order m are exactly the integral holomorphic curves of the
complex Pfaffian system ⊕m
j=1[mj] ⊂ TN in the almost complex manifold (N, J).
0.2. The determined case.
We call “the minimal determined system” the determined system of minimal or-
der mk , and “the maximal determined system” the determined system of maximal
order k
− 1.
Any solution of a determined system is solution of the corresponding maximal
determined system. More precisely, a map f : L → G/G0 is solution of a deter-
mined system (associated to G/G0) if and only if it is solution of the maximal
determined system (associated to G/G0) and satisfies an additional holomorphicity
condition. When this holomorphicity condition is maximal, then we obtain the
minimal determined system.
0.2.1. The minimal determined system. The minimal determined system has an
interpretation in terms of horizontally holomorphic and vertically harmonic maps
f : L → N = G/G0. It also has an equivalent interpretation in terms of vertically
harmonic twistor lifts in some twistor space. Let us make precise this point.
In the even case. As we have seen in the subsection 0.1 below, the homoge-
neous space N = G/G0 admits a G-invariant splitting TN = H ⊕ V corresponding
to the fibration π: N → M and N is naturally endowed with an f-structure F
which defines a complex structure on the horizontal subbundle H and vanishes
on the vertical subbundle V. Then we say that a map f : L → N is horizontally
holomorphic if
(df ◦ jL)H
= F ◦ df.
Then we prove that the even minimal determined system (Syst(k, τ)) means that
the geometric map f is horizontally holomorphic and vertically harmonic, i.e.
τv
(f) := Trg(∇v
dv
f) = 0
(for any hermitian metric g on the Riemann surface L). Here ∇v
is the vertical
component of the Levi-Civita connection ∇ (of some G-invariant metric on N).
Moreover, vertically harmonic maps are exactly the critical points (w.r.t. vertical](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-17-320.jpg)
![INTRODUCTION 3
variations) of the vertical energy functional:
Ev
(u) =
1
2
L
|dv
u|2
dvolg.
We prove also that this system also has an equivalent interpretation in terms
of vertically harmonic twistor lifts in some twistor space Z2k,j(M, J2) which is a
subbundle of Z2k(M), where
Zk (M) = {J ∈ SO(TM)| Jk
= Id, Jp
= Id if p k
, ker(J ± Id) = {0}}
is the bundle of isometric endomorphisms of TM with finite order k
and with no
eigenvalues = ±1. More precisely denoting by J2 the section of SO(TM), of order
k, defined by τ2
m, then we define Z2k,j(M, J2) = {J ∈ Z2k(M)|J2
= J2}. Then
we prove that N = G/G0 can be embedded into the twistor space Z2k,j(M, J2)
via a natural morphism of bundle over M = G/H. We prove that f : L → N is
solution of the system if and only if the corresponding map Jf
: L → Z2k,j(M, J2)
is a vertically harmonic twistor lift.
In the odd case. We obtain an analogous interpretation as in the even case.
An interpretation in terms of horizontally holomorphic and vertically harmonic
maps f : L → N = G/G0. Let us make precise that in the odd case, the action
functional in the variational interpretation has a Wess-Zumino term in addition to
the vertical energy (See below in this introduction).
Moreover by embedding G/G0 into the twistor space Z2k+1(N) of order 2k + 1
isometric endomorphisms in TN, we obtain an interpretation in terms of vertically
harmonic twistor lift.
0.2.2. The general structure of the maximal determined case. First, the maxi-
mal determined system has 3 model cases. This means that we can distinguish 3
maximal determined systems, namely the three maximal determined systems with
lowest order of symmetry (2,3,4). Their corresponding geometric equations (when
put all together) contain already all the structure terms - in a simple form- that
will appear in the further maximal determined systems in a more complex and
general form due to the more complex geometric structure in the further maximal
determined systems. That is in this sense that we can say that all the further
determined systems associated to target spaces N with higher order of symmetry
will be modeled on these model systems.
0.2.3. The model system in the even case. In the even case, this model is the
first elliptic integrable system associated to a symmetric space (m = 1, k
= 2)
which is - as it is well known - exactly the equation of harmonic maps from the
Riemann surface L into the symmetric space under consideration. This is the
“smallest” determined system, i.e. with lowest order of symmetry in the target
space N = G/G0. In this case -N is symmetric- the determined case is reduced to
one system, the one of order 1.
0.2.4. The model system in the odd case. In the odd case, this model is the
second elliptic integrable system associated to a 3-symmetric space. This is the
“smallest” determined system in the odd case, i.e. with lowest odd order of sym-
metry in the target space N = G/G0. We prove that this system is exactly the
equation for holomorphically harmonic maps into the almost complex manifold
(N, J) with respect to the anticanonical connection ∇1
= ∇0
+ [ , ][m], where ∇0](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-18-320.jpg)
![4 IDRISSE KHEMAR
is the canonical connection. Or equivalently this is the equation for holomorphi-
cally harmonic maps into the almost complex manifold (N, −J) with respect to the
canonical connection ∇0
.
Holomorphically harmonic maps. Given a general almost complex mani-
fold (N, J) with a connection ∇, we define holomorphically harmonic maps f : L →
N as the solutions of the equation
(0.1)
¯
∂∇
∂f
1,0
= 0
where [ ]1,0
denotes the (1, 0)-component according to the splitting TNC
= T1,0
N ⊕
T0,1
N defined by J. This equation is equivalent to
d∇
df + Jd∇
∗ df = 0
or, equivalently, using any Hermitian metric g on L
Tg(f) + Jτg(f) = 0
where Tg(f) = ∗f∗
T = f∗
T(e1, e2), with (e1, e2) an orthonormal basis of TL, and
τg(f) = ∗d∇
∗ df = Trg(∇df) is the tension field of f. Of course Trg denotes the
trace with respect to g, and the expression ∇df denotes the covariant derivative of
df with respect to the connection induced in T∗
L⊗f∗
TN by ∇ and the Levi-Civita
connection in L.
In particular, we see that if ∇ is torsion free or more generally if f is torsion
free, i.e. f∗
T = 0, then holomorphic harmonicity is equivalent to (affine) harmonic-
ity. Therefore, this new notion is interesting only in the case of a nontorsion free
connection ∇.
The vanishing of some ¯
∂∂-derivative. Now, let us suppose that the con-
nection ∇ on N is almost complex, i.e. ∇J = 0. Then, according to equation (0.1),
we see that any holomorphic curve f : (L, jL) → (N, J) is anti-holomorphically
harmonic, i.e. holomorphically harmonic with respect to −J. In particular, this
allows to recover that a 1-primitive solution (i.e. of order m = 1) of the elliptic sys-
tem associated to a 3-symmetric space is also solution of the second elliptic system
associated to this space.
Moreover, the holomorphically harmonic maps admit a formulation very anal-
ogous to that of harmonic maps in terms of the vanishing of some ¯
∂∂-derivative,
which implies a well kown characterisation in terms of holomorphic 1-forms. Indeed
we prove that f : (L, jL) → (N, J, ∇) is holomorphically harmonic if and only if
(0.2) ˆ
∂
∇
ˆ
∂f = 0,
i.e. ˆ
∂f is a holomorphic section of T∗
1,0L⊗C f∗
TN. Here the hat “ˆ” means that we
extend a 1-form on TL, like d or ∇, by C-linearity as a linear map from TLC
into
the complex bundle (TN, J). In other words instead of extending these 1-forms
as C-linear maps from TLC
into TNC
as it is usual, we use the already existing
structure of complex vector bundle in (TN, J) and extend these very naturally as C-
linear map from TLC
into the complex bundle (TN, J). Therefore we can conclude
that holomorphically harmonic maps have the same formulation as harmonic maps
with the difference that instead of working in the complex vector bundle TNC
, we
stay in TN which is already a complex vector bundle in which we work.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-19-320.jpg)
![INTRODUCTION 5
The sigma model with a Wess-Zumino term. Finally, let us suppose that
N is endowed with a ∇-parallel Hermitian metric h. Therefore (N, J, h) is an
almost Hermitian manifold with a Hermitian connection ∇. Suppose also that J
anticommutes with the torsion T of ∇ i.e.
T(X, JY ) = −JT(X, Y )
which is equivalent to
T =
1
4
NJ
where NJ denotes the torsion of J i.e its Nijenhuis tensor.
Suppose also that the torsion of ∇ is totally skew-symmetric i.e. the trilinear
map
T∗
(X, Y, Z) = T(X, Y ), Z
is a 3-form. Lastly, we suppose that the torsion is ∇-parallel, i.e. ∇T∗
= 0 which
is equivalent to ∇T = 0. Then we prove that this implies that the 3-form
H(X, Y, Z) = −T∗
(X, Y, JZ) = JT(X, Y ), Z
is closed dH = 0.2
Then the equation for holomorphically harmonic maps f : L → N is the equa-
tion of motion (i.e. Euler-Lagrange equation) for the sigma model in N with the
Wess-Zumino term defined by the closed 3-form H. The action functional is given
by
S(f) = E(f) + SW Z
(f) =
1
2
L
|df|2
dvolg +
B
H,
where B is 3-submanifold (or indeed a 3-chain) in N whose boundary is f(L).
Then since dH = 0, the variation of the Wess-Zumino term is a boundary term
δSW Z
=
B
Lδf H =
B
dıδf H =
f(L)
ıδf H,
whence its contribution to the Euler-Lagrange equation involves only the original
map f : L → N.
In particular, applying this result to the case we are interested in, i.e. N is
3-symmetric, we obtain:
The second elliptic system associated to a 3-symmetric space N = G/G0 is the
equation of motion for the sigma model in N with the Wess-Zumino term defined
by the closed 3-form H(X, Y, Z) := T∗
(X, Y, JZ), where T is the torsion of the
canonical connection ∇0
and J is the canonical almost complex structure.3
2Let us point out that in general T∗ is not closed even if it is ∇-parallel. For example,
in a Riemannian naturally reductive homogeneous space G/H, endowed with its canonical con-
nection ∇0, we have ∇0T = 0 but dT∗(X, Y, Z, V ) = −2Jacm(X, Y, Z), V where Jacm is the
m-component of the Jacobi identity (i.e. the sum of the circular permutations of [X, [Y, Z]m]m).
Of course m denotes the AdH-invariant summand in the reductive decomposition g = h ⊕ m.
3In fact, we need a naturally reductive metric on N to ensure that T∗ is a 3-form. But if we
allow Pseudo-Riemannian metrics and if g is semisimple then the metric defined by the Killing
form is naturally reductive. In fact, the elliptic integrable system is a priori written in an affine
context, i.e. its natural - in the sense of initial- geometric interpretation takes place in the context
of affine geometry in terms of the linear connections ∇t = ∇0 + t[ , ][m]. If we want that this
interpretation takes place in the context of Riemannian geometry we need, of course, to add some
hypothesis of compactness, like the compactness of AdmG0 and the natural reductivity. But we
do not need these hypothesis if we work in the Pseudo-Riemannian context.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-20-320.jpg)
![8 IDRISSE KHEMAR
Best geometric setting. It turns out that the more rich geometric context in
which stringy harmonicity admits interesting properties is the one of G1-manifolds,
more precisely G1-manifolds whose the characteristic connection has a parallel tor-
sion. Making systematic use of the covariant derivative of the Kähler form, A.
Gray and L. M. Hervella, in the late seventies, classified almost Hermitian struc-
tures into sixteen classes [27]. Denote by W the space of all trilinear forms (on
some Hermitian vector space, say Ty0
N for some reference point y0 ∈ N) having
the same algebraic properties as ∇h
ΩJ . Then they proved that we have a U(n)-
irreducible decomposition W = W1 ⊕ W2 ⊕ W3 ⊕ W4. The sixteen classes are then
respectively the classes of almost Hermitian manifolds for which ∇h
ΩJ ‘lies in’ the
U(n)-invariant subspaces WI = ⊕i∈I Wi, I ⊂ {1, . . . , 4}, respectively. In particular,
if we take as invariant subspace {0}, we obtain the Kähler manifolds, if we take W1,
we obtain the class of nearly Kähler manifolds. Moreover the class of G1-manifolds
is the one defined by G1 = W1 ⊕ W3 ⊕ W4. It is characterised by : (N, J, h) is
of type G1 if and only if the Nijenhuis tensor NJ is totally skew-symmetric (i.e. a
3-form).
In this paper, we prove the following theorem4
:
Theorem 0.2.2. An almost Hermitian manifold (N, J, h) admits a Hermitian
connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor
NJ is itself totally skew-symmetric. In this case, the connection is unique and
determined by its torsion which is given by
T = −dc
ΩJ + NJ .
The characteristic connection is then given by ∇ = ∇h
− 1
2 T.
Then we prove:
Proposition 0.2.1. Let us suppose that the almost Hermitian manifold (N,J, h)
is a G1-manifold. Let us suppose that its characteristic connection ∇ has a parallel
torsion ∇T = 0. Then the 3-form
H(X, Y, Z) = T(JX, JY, JZ) = (J · T)(X, Y ), Z
is closed dH = 0.
This then gives us the following variational interpretation.
Theorem 0.2.3. Let us suppose that the almost Hermitian manifold (N, J, h)
is a G1-manifold. Let us suppose that its characteristic connection ∇ has a parallel
torsion ∇T = 0.
Then the equation for stringy harmonic maps f : L → N is exactly the Euler-
Lagrange equation for the sigma model in N with a Wess-Zumino term defined by
the closed 3-form
H = −dΩJ + JNJ .
Moreover any (2k + 1)-symmetric space (G/G0, J, h) endowed with its canon-
ical almost complex structure and a naturally reductive G-invariant metric h (for
which J is orthogonal) is a G1-manifold and moreover its characteristic connection
coincides with its canonical connection ∇0
. Finally, the torsion of the canonical
4After the fact, we realized that it has already been proved by Friedrich-Ivanov[24]. However
we give a different and completely written proof. See remark 5.3.4](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-23-320.jpg)
![INTRODUCTION 11
characteristic connection. Furthermore, the torsion of the canonical connection is
obviously parallel. Finally, we prove that any 2k-symmetric space (G/G0, F, h)
satisfies the two hypothesis above. Therefore we obtain an interpretation of the
maximal determined even system associated to a 2k-symmetric space in terms of a
sigma model with a Wess-Zumino term.
A particular case: Horizontally Kähler f-manifolds. Let (N, F, h) be a
metric f-manifold. We will say that (N, F, h) is horizontally Kähler if DF|H3 = 0,
where D is the Levi-Civita connection of h.
Then, we prove that in this case, the two hypothesis above (which characterise
the closure of the 3-form H) are satisfied. Moreover, any characteristic connection
∇ satisfies TH3 = 0, which leads to special properties.
A example of this situation is given by any 4-symmetric space, endowed with
its canonical f-structure and a naturally reductive G-invariant metric (compatible
with F).
0.2.8. The intermediate determined systems. For the intermediate determined
systems (mk m k
− 1), these are obtained from the maximal determined
case by adding holomorphicity in the subbundle Hm
= ⊕
m
j=1[mj] ⊂ H, where
m = k
− 1 − m. It means that the m-th determined system has a geometric
interpretation in terms of stringy harmonic maps which are Hm
-holomorphic:
(df ◦ jL)Hm
= F[m]
◦ df.
We have seen that the maximal determined system has an interpretation in terms
of a sigma model with a Wess-Zumino term defined by a 3-form H. In fact, more
generally, let mk ≤ m ≤ k
− 1, and let us consider the splitting TN = Hm
⊕ Vm
defined above. Then one can prove that any m-th determined system is the Euler-
Lagrange equation w.r.t. vertical variations (i.e. in Vm
) of the following functional
Ev̄
(f) =
1
2
L
|dv̄
f|2
dvolg +
B
Hv̄
where dv̄
f = [df]Vm
, Hv̄
= H − H = H|S(Vm,Hm), H = H|(Hm)3 , and B is a 3-
submanifold of N with boundary ∂B = f(L). We have set S(Vm
, Hm
) = S(Vm
×
Hm
× Hm
) and S(E1 × E2 × E3) = S
i,j,k
Ei ⊗ Ej ⊗ Ek for any E1, E2, E3 vector
bundles over N.
0.3. The underdetermined case.
We prove that the m-th underdetermined system (m k
− 1) is in fact equiv-
alent to some m-th determined or primitive system associated to some new auto-
morphism τ̃ defined in a product gq+1
, of the initial Lie algebra g. More precisely,
we write
m = qk
+ r, 0 ≤ r ≤ k
− 1
the Euclidean division of m by k
. Then we consider the automorphism in gq+1
defined by
τ̃ : (a0, a1, . . . , aq) ∈ gq+1
−→ (a1, . . . , aq, τ(a0)) ∈ gq+1
.
Then τ̃ is of order (q + 1)k
. We prove that the initial m-th (underdetermined)
system associated to (g, τ) is in fact equivalent to the m-th (determined) system
associated to (gq+1
, τ̃).](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-26-320.jpg)
![12 IDRISSE KHEMAR
0.4. In the twistor space.
For each previous geometric interpretation in the target space N = G/G0, there
is a corresponding geometric interpretation in the twistor space.
• These previous geometric interpretations take place in some manifolds en-
dowed with some particular structure. This could simply be, for example, a struc-
ture of almost complex manifold in the case of the interpretation of the stringy
harmonicity in terms of the vanishing of some ¯
∂∂-derivative but it could also be
the more strong structure of G1-manifolds whose characteristic connection has a
parallel torsion. Moreover, our k
-symmetric spaces are very particular examples
of this kind of manifolds. Therefore it is natural to try to make these interpreta-
tions more universal by writting them in a more general setting. More precisely we
would like to find some universal prototype of these ”special” manifolds, which can
be endowed canonically with the needed geometric structure and such that any of
our special manifolds can be embedded in this prototype (see also [48]).
Indeed, as concerns k
-symmetric spaces, we know that they can be embed-
ded canonically into some twistor bundles. In the even case we have an injective
morphim of bundle over M = G/H defined by the embedding
G/G0 → Z2k(G/H),
whereas in the odd case we have a chapter defined by the embedding
G/G0 → Z2k+1(G/G0).
• In the even case, the fibration π: G/G0 → G/H imposes to view canonically any
2k-symmetric space as a subbundle of Z2k(G/H) so that the twistorial interpre-
tation is in some sense dictated by the structure of the 2k-symmetric space. The
geometric interpretations in the twistor spaces are universal since these twistor
spaces are defined for any Riemannian manifold and are endowed canonically with
the different geometric structures that we need. That is to say the geometric
structures we need to endow the target space N with, in our previous geometric
interpretations.
More generally, suppose that we want to study stringy harmonicity in metric
f-manifolds (N, F, h). It is then natural to consider the particular case where the
vertical subbundle V is the tangent space to the fibre of a Riemannian submersion
π: (N, h) → (M, g), i.e. V = ker dπ. For example, we have seen that among the
sufficient list of conditions for our variationnal interpretation of stringy harmonicity
there is the condition RV = 0. In the particular case of a Riemannian submersion,
the f-structure F defines a complex structure ¯
J on π∗
TM = H which itself gives
rise to a morphism of submersion I : N → Σ(M), y → (π(y), ¯
J(y)). This shows that
the twistor bundle Σ(M) appears naturally in the general context - even though
the morphism I is not injective in general.
Furthermore, an interesting class of Riemannian submersion π: (N, h) → (M, g)
is the one of homogeneous fibre bundles, of which the twistor bundles Zp(M) are
particular examples (p ∈ N∗
). For example, vertically harmonic sections of homo-
geneous fibre bundles have been investigated by C.M. Wood [71, 72].](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-27-320.jpg)
![INTRODUCTION 13
0.5. Related subjects and works, and motivations.
0.5.1. Relations with surface theory. The theory of harmonic maps of surfaces
has been greatly enriched by ideas and methods from integrable systems [12, 16,
14, 18, 23, 31, 32, 67]. In particular, these ideas have revolutionised the theory
of harmonic maps from a surface into a symmetric space and so, via an appropriate
Gauss map, the theory of constant mean curvature surfaces and Willmore surfaces
among others. For example, Pinkall and Sterling [59] were able to give an algebro-
geometric construction of all constant mean curvature tori while Dorfmeister-Pedit-
Wu [23] gave a Weierstrass formula for all constant mean curvature immersions of
any (simply connected) surface in terms of holomorphic data. These advances
were taken up by Hélein and Romon [33, 34, 35] who showed that similar ideas
could be applied to the study of Hamiltonian stationary Lagrangian surfaces in a 4-
dimensional Hermitian symmetric space. It was the first example of second elliptic
integrable system associated to a 4-symmetric space. In [42], we presented a new
class of isotropic surfaces in the Euclidean space of dimension 8 by identifying R8
with the set of octonions O, and we proved that these surfaces are solutions of a
second elliptic integrable system associated to a 4-symmetric space. By restriction
to R4
= H, we obtained the Hamiltonian stationary Lagrangian surfaces and by
restriction to R3
= Im H, we obtained the CMC surfaces. Furthermore, in [44],
we presented a geometric interpretation of all the second elliptic integrable systems
associated to a 4-symmetric space in terms of vertically harmonic twistor lifts of
conformal immersions into the Riemannian symmetric space (associated to our 4-
symmetric space) (see also [17]). When the previous Riemannian symmetric space
is 4-dimensional, then any conformal immersion admits an unique twistor lift and
the vertical harmonicity of this twistor lift is equivalent to the holomorphicity of
the mean curvature vector of the conformal immersion (see [17]). In particular,
when the Riemannian symmetric space is Hermitian, one obtains a conceptual
explanation of the result of Hélein-Romon.
0.5.2. Relations with mathematical physics.
Metric connections with totally skew-symmetric torsion. We refer the
reader to [2, 24] about connections with skew-symmetric torsions and their relations
to physics and more particulary string theory. Linear metric connections with
totally skew-symmetric torsion recently became a subject of interest in theoretical
and mathematical physics. Let us give here some examples (taken from [24]).
• The target space of supersymmetric sigma models with Wess-Zumino term
carries a geometry of a metric connection with skew-symmetric torsion.
• In supergravity theories, the geometry of the moduli space of a class of black
holes is carried out by a metric connection with skew-symmetric torsion.
• The geometry of NS-5 brane solutions of type II supergravity theories is
generated by a metric connection with skew-symmetric torsion.
• The existence of parallel spinors with respect to a metric connection with
skew-symmetric torsion on a Riemannian spin manifold is of importance in string
theory, since they are associated with some string solitons (BPS solitons).
The sigma-models. The classical non-linear sigma model describes harmonic
maps between two (pseudo)riemannian manifolds, which are called the spacetime
and the target space. Nonlinear sigma-models provide a much-studied class of field
theories of both phenomenological and theoretical interest. The chiral model for](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-28-320.jpg)
![14 IDRISSE KHEMAR
example summarizes many low energy QCD interactions while 2-dimensional sigma-
models may possess nontrivial classical field configurations and have analogies with
4-dimensional Yang-Mills equations but are simpler to handle. The study of Wess-
Zumino (WZ) terms has received considerable attention since they were introduced
in four-dimensional chiral field theories as effective Lagrangians describing the low
energy consequences of the anomalous Ward identities of the theory . Sigma-models
also have connections with string theories.
Two dimensional sigma-models have already proven a fertile arena for the inter-
play of topology, geometry, and physics. The supersymmetric sigma-models have
been generalized by introducing a Wess-Zumino term into the Lagrangian. This
term may be interpreted as adding torsion to the canonical Levi-Civita connection
of the earlier models. The addition of such torsion can impose constraints on the
possible geometries of the target [13].
Our contribution. We give new examples of integrable two-dimensional non-
linear sigma models. These new examples take place in some homogeneous spaces,
namely k
-symmetric spaces, which are not symmetric spaces. At our knowledge,
all the already known integrable two-dimensional nonlinear sigma models take place
in symmetric spaces or (equivalently) in Lie groups.
0.5.3. Relations of F-stringy harmonicity and supersymmetry. The P.D.E of
F-stringy harmonicity splits following the splitting TN = H ⊕ V defined by the
f-structure F.
More precisely, this equation is a coupling between the equation of ¯
J-stringy
harmonicity and the equation of vertical harmonicity, the coupling terms calling out
the curvatures of H and V, and the component NF |V×H of the Nijenhuis tensor.
Moreover, let us suppose that we have we have a fibration π: N → M. Then
we have a supersymmetric interpretation of the F-stringy harmonicity: F-stringy
harmonicity can be viewed as a supersymmetric extension of the J-stringy har-
monicity. In the splitting TN = H ⊕ V, the horizontal subbundle played the role
of the odd part and the vertical subbundle plays the role of the even part. In other
words, the bosonic equation is a harmonic map equation (the vertical harmonicity)
and the fermionic equation is the ¯
J-stringy harmonic map equation.
Let us also mention that, in [43], we obtained a supersymmetric interpretation
of all the second elliptic integrable systems associated to a 4-symmetric space in
terms of superharmonic maps.
Acknowledgements. The author wishes to thank Josef Dorfmeister for his useful
comments on the first parts of this paper. He is also grateful to him for his interest
in the present work, his encouragements as well as his support throughout the
preparation of this paper.
This work started at T.U. Munich and was completed at Universität Tübingen.
The author would like to thank these two institutions. He also wishes to thank
Franz Pedit for allowing him to finish this work in very good conditions and an
excellent environment at the Mathematisches Institut of Tübingen.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-29-320.jpg)
![Notation, conventions and general definitions
0.6. List of notational conventions and organisation of the paper.
• Let k ∈ N∗
. Then we will often confuse - when it is convenient to do it- an
element in Zk with one of its representants. For example, let (ai)i∈Zk
be a family
of elements in some vector space E, and 0 ≤ m k/2 an integer. Then we will
write
ai = a−i 1 ≤ i ≤ m
to say that this equality holds for all i ∈ {1 + kZ, . . . , m + kZ} ⊂ Zk.
• Let us suppose that a vector space E admits some decomposition E = ⊕i∈I Ei.
Then, for any vector v ∈ E we denote by [v]Ei
its component in Ei. It could also
happen sometimes that we write this component [v]Ei
.
• We denote EC
:= E ⊗ C the complexification of a real vector space.
• Furthermore let A ∈ End(E) be an endomorphism of a finite dimensional real
vector space that we suppose to be diagonalizable. Then if its complex spectrum is
{λi, i ∈ I}, we will write its eigenspace decomposition in the form EC
= ⊕i∈I EC
i ,
where EC
i = ker(A − λiId). Moreover, if λi ∈ R for some i ∈ I, then we set
Ei := EC
i ∩ E so that EC
i = (Ei)C
, for these particular i ∈ I.
• We denote by ωp a p-th primitive root of unity, which will be often chosen
equal to e2iπ/p
.
• We denote by C∞
(M, N) the set of smooth maps from a manifold M into
a manifold N. Now, let π: N → M be a surjection, then we denote by C(π) the
set of sections of π, i.e. the maps s such that π ◦ s = IdM . If there is no risk of
confusion we will also use the notation C(N). Furthermore, let p: E → M be a
vector bundle. Then we will denote by the same letter p its tensorial extensions:
p: End(E) → M and so on.
• Let (M, g) be a Riemannian manifold. Then we denote by ∗ its Hodge opera-
tor (and ∗g if the metric need to be precised). Let (E, h) → M be a Riemannian vec-
tor bundle over a manifold M, we denote by Σ(E) the bundle of orthogonal almost
complex structures in E. In particular if E = TM then we set Σ(M) := Σ(TM).
• More generally, let Z(Rn
) ⊂ End(Rn
) (resp. O(n)) be some submanifold
defined for any n ∈ N∗
. This allows us to define Z(E) for any (Riemannian) vector
bundle E and we set in particular Z(M) = Z(TM) for any (Riemannian) manifold.
• Moreover it will happen that we will write ”SO(E)” without precising that
the Riemannian vector bundle E is supposed to be oriented. We will consider that
this hypothesis is implicit once we write this kind of symbol.
• Let (A, +, ×) be an associative K-algebra over the field K. Then for any
a ∈ A, we set Com(a) = {b ∈ A|ab = ba} and Ant(a) = {b ∈ A|ab = −ba}.
• w.r.t. : with respect to.
A list of notation and an index are available at the end of the paper. Bibli-
ographic remarks and a summary of our own contributions are
15](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-30-320.jpg)
![16 NOTATION, CONVENTIONS AND GENERAL DEFINITIONS
available at the end of each chapter from chapter 2 to section 7. The numbering
of the equations is made by section: e.g. equation (2.13) is the 13th equation of
chapter 2. Moreover the numbering of theorems, definitions and such is made by
subsection: e.g. Theorem 2.2.1 is the first theorem in subsection 2.2 (but it is not
in subsubsection 2.2.1).
0.7. Almost complex geometry.
Let E be a real vector space endowed with a complex structure: J ∈ End(E),
J2
= −Id. Then we denote by E1,0
and E0,1
respectively the eigenspaces of J
associated to the eigenvalues ±i respectively. Then we have the following eigenspace
decomposition
(0.3) EC
= E1,0
⊕ E0,1
and the following equalities
(0.4)
E1,0
= ker(J − iId) = (J + iId)EC
E0,1
= ker(J + iId) = (J − iId)EC
so that remarking that (J ± iId)iE = (Id ∓ iJ)E = (Id ∓ iJ)JE = (J ± iId)E, we
can also write
(0.5)
E1,0
= (J + iId)E = (Id − iJ)E = {X − iJX, X ∈ E}
E0,1
= (J − iId)E = (Id + iJ)E = {X + iJX, X ∈ E}
In the same way we denote by
(E∗
)C
= E∗
1,0 ⊕ E∗
0,1
the decomposition induced on the dual E∗
by the complex structure J∗
: η ∈ E∗
→
ηJ ∈ E∗
. Besides, given a vector Z ∈ EC
, we denote by
Z = [Z]1,0
+ [Z]0,1
its decomposition according to (0.3). Let us remark that
[Z]1,0
= (Id − iJ)Z and [Z]0,1
= (Id + iJ)Z.
Moreover, given η a n-form on E, we denote by η(p,q)
its component in Λp,q
E∗
according to the decomposition
Λn
E∗
= ⊕p+q=nΛp,q
E∗
,
where Λp,q
E∗
= Λp
E∗
1,0 ∧ Λq
E∗
0,1 . However for 1-forms, we will often prefer the
notation η = η
+ η
, where η
and η
denote respectively η(1,0)
and η(0,1)
.
More generally, all what precedes holds naturally when E is a real vector bundle
over a manifold M, endowed with a complex structure J.
We will write
d = ∂ + ¯
∂
the decomposition of the exterior derivative of differential forms on an almost com-
plex manifold (M, J), according to the decomposition TMC
= T1,0
M ⊕ T0,1
M.
We will denote by Hol((M, JM
), (N, JN
)) := {f ∈ C∞
(M, N)| df ◦ JM
= JN
◦
df} the set of holomorphic maps between two almost complex manifolds (M, JM
)
and (N, JN
).
In this paper, we will use the following definitions.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-31-320.jpg)
![NOTATION, CONVENTIONS AND GENERAL DEFINITIONS 17
Definition 0.7.1. Let E be a real vector bundle. An f-structure in E is an
endomorphism F ∈ C(EndE) such that F3
+ F = 0. An f-structure on a manifold
M is an f-structure in TM. A manifold (M, F) endowed with an f-structure is
called an f-manifold.
An f-structure F in a vector bundle E is determined by its eigenspaces decom-
position that we will denote by
EC
= E+
⊕ E−
⊕ E0
where E±
= ker(F ∓ iId) and E0
= ker F. In particular if E = TM, then we will
set Ti
M = (TM)i
, ∀i ∈ {0, ±1}.
Definition 0.7.2. Let (M, FM
) and (N, FN
) be f-manifolds. Then a map
f : (M, FM
) → (N, FN
) is said to be f-holomorphic if it satisfies
df ◦ FM
= FN
◦ df
Definition 0.7.3. Let (M, JM
) be an almost complex manifold and N a man-
ifold with a splliting TN = H ⊕ V. Let us suppose that the subbundle H is
naturally endowed with a complex structure JH
. Then we will say that a map
f : (M, JM
) → N is H-holomorphic if it satisfies the equation
[df]H
◦ JM
= JH
◦ [df]H
,
where [df]H
is the projection of df on H along V. Moreover, if for some reason, H
inherits the name of horizontal subbundle, then we will say that f is horizontally
holomorphic.
Remark 0.7.1. Let us remark that an f-structure in a manifold N is equivalent
to a splitting TN = V ⊕ H together with a complex structure JH
on H.
Definition 0.7.4. An affine manifold (N, ∇) is a manifold endowed with a
linear connection.
An almost complex affine manifold (N, J, ∇) is an almost complex manifold
endowed with an almost complex connection: ∇J = 0.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-32-320.jpg)
![CHAPTER 1
Invariant connections on reductive
homogeneous spaces
The references for this chapter where we recall some results that we will need
in this paper, are [54], [61], [20], and to a lesser extent [1] and [37].
1.1. Linear isotropy representation.
Let M = G/H be a homogeneous space with G a real Lie group and H a
closed subgroup of G. G acts transitively on M in a natural manner which defines
a natural representation: φ: g ∈ G → (φg : x ∈ M → g.x) ∈ Diff(M). Then ker φ
is the maximal normal subgroup of G contained in H. Further, let us consider the
linear isotropy representation:
ρ: h ∈ H → dφh(x0) ∈ GL(Tx0
M)
where x0 = 1.H is the reference point in M. Then we have ker ρ ⊃ ker φ. Moreover
the linear isotropy representation is faithful (i.e. ρ is injective) if and only if G acts
freely on the bundle of linear frames L(M).
We can always suppose without loss of generality that the action of G on M
is effective (i.e. ker φ = {1}) but it does not imply in general that the linear
isotropy representation is faithful. However if there exists on M a G-invariant
linear connection, then the linear isotropy representation is faithful provided that
G acts effectively on M. (Indeed, given a manifold M with a linear connection,
and x ∈ M, an affine transformation f of M is determined by (f(x), df(x)), i.e. f
is the identity if and only if it leaves one linear frame fixed.)
1.2. Reductive homogeneous space.
Let us suppose now that G/H is reductive, i.e. there exists a decomposition
g = h⊕m such that m is AdH-invariant: ∀h ∈ H, Adh(m) = m. Then the surjective
map ξ ∈ g → ξ.x0 ∈ Tx0
M has h as kernel and so its restriction to m is an
isomorphism m ∼
= Tx0
M. This provides an isomorphism of the associated bundle
G ×H m with TM by:
(1.1) [g, ξ] → g.(ξ.x0) = Adg(ξ).x
where x = π(g) = g.x0.
Moreover, we have a natural inclusion G ×H m → G ×H g and the associated
bundle G ×H g is canonically identified with the trivial bundle M × g via
(1.2) [g, ξ] → (π(g), Adg(ξ)).
Thus we have an identification of TM with a subbundle [m] of M × g, which we
may view as a g-valued 1-form β on M given by:
βx(ξ.x) = Adg[Adg−1
(ξ)]m,
19](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-34-320.jpg)
![20 IDRISSE KHEMAR
where π(g) = x, ξ ∈ g and [ ]m is the projection on m along h. Equivalently, for all
X ∈ TxM, β(X) is the unique element ξ ∈ [m]x (= Adg(m), with π(g) = x) such
that X = ξ.x, in other words β(X) is characterized by
β(X) ∈ [m]x ⊂ g and X = β(X).x .
In fact, β is nothing but the projection on M of the H-equivariant 1-form, θm, on
G, i.e. θm is the H-equivariant lift of β. Here, θm is defined as the m-component of
the left invariant Maurer-Cartan form θ of G. This can be written as follows
(1.3) (π∗
β)g = Adg(θm) ∀g ∈ G
with θg(ξg) = g−1
.ξg for all g ∈ G, ξg ∈ TgG.
Notation. For any AdH-invariant subspace l ⊂ m, we will denote by [l] the
subundle of [m] ⊂ M × g defined by [l]g.x0
= Adg(l).
1.3. The (canonical) invariant connection.
On a reductive homogeneous space M = G/H, the Ad(H)-invariant summand
m provides by left translation in G, a G-invariant distribution H(m), given by
H(m)g = g.m which is horizontal for π: G → M and right H-invariant and thus
defines a G-invariant connection in the principal bundle π: G → M. In fact this
procedure defines a bijective correspondance between reductive summands m and
G-invariant connections in π: G → M (see [54], chap. 2, Th 11.1). Then the
corresponding h-valued connection 1-form ω on G (of this G-invariant connection)
is the h-component of the left invariant Maurer-Cartan form of G:
ω = θh.
1.4. Associated covariant derivative.
The connection 1-form ω induces a covariant derivative in the associated bundle
G ×H m ∼
= TM and thus a G-invariant covariant derivative ∇0
in the tangent
bundle TM. In particular, we can conclude according to section 1.1 that if G/H is
reductive then the linear isotropy representation is faithful (provided that G acts
effectively) or equivalently that ker Adm = ker ρ = ker φ. We will suppose in the
following that, without explicit reference to the contrary, the action of G is effective
and (thus) the linear isotropy representation is faithful.
One can compute explicitly ∇0
.
Lemma 1.4.1. [20]
β(∇0
XY ) = X.β(Y ) − [β(X), β(Y )], X, Y ∈ Γ(TM).
Let us write (locally) β(X) = AdU(Xm), β(Y ) = AdU(Ym) where U is a (local)
section of π and Xm, Ym ∈ C∞
(M, m) then we have (using the previous lemma)
β(∇0
XY ) = AdU (dYm(X) + [α(X), Ym] − [Xm, Ym])
= AdU (dYm(X) + [αh(X), Ym] + [αm(X) − Xm, Ym])
where α = U−1
.dU. Besides since U is a section of π (π ◦ U = Id), then pulling
back (1.3) by U, we obtain β = AdU(αm) and then αm(X) = Xm, so that
(1.4) β(∇0
XY ) = AdU (dYm(X) + [αh(X), Ym])
Remark 1.4.1. We could also say that Xm, Ym are respectively the pullback
by U of the H-equivariant lifts X̃, Ỹ of X, Y (given by β(Xπ(g)) = Adg(X̃(g))).](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-35-320.jpg)
![1. INVARIANT CONNECTIONS ON REDUCTIVE HOMOGENEOUS SPACES 21
Then ∇0
X Y lifts as the m-valued H-equivariant map on G:
∇0
XY = dỸ (X̃) + [θh(X̃), Ỹ ]
and then taking the U-pullback we obtain the previous result (without using
lemma 1.4.1).
Moreover, we can express ∇0
in terms of the flat differentiation in the trivial
bundle M × g (⊃ [m]). Let us differentiate the equation Y = AdU(Ym) (we use the
identification TM = [m] ⊂ M × g)
dY =AdU (dYm +[α, Ym])=AdU (dYm + [αh, Ym])+AdU (([αm, Ym]))=∇0
Y+[β, Y ].
Finally, we obtain
(1.5) dY = ∇0
Y + [β, Y ]
and we recover lemma 1.4.1.
1.5. G-invariant linear connections in terms of equivariant bilinear
maps.
Now let us recall the following results about invariant connections on reductive
homogeneous spaces.
Theorem 1.5.1. [54] Let πP : P → M be a K-principal bundle over the re-
ductive homogeneous space M = G/H and suppose that G acts on P as a group of
automorphisms and let u0 ∈ P be a fixed point in the fibre of x0 ∈ M (πP (u0) = x0).
There is a bijective correspondance between the set of G-invariant connections ω in
P and the set of linear maps Λm : m → k such that
(1.6) Λm(hXh−1
) = λ(h)Λm(X)λ(h)−1
for X ∈ m and h ∈ H
where λ: H → K is the morphism defined by hu0 = u0λ(h) (H stabilizes the fibre
Px0
= u0.K). The correspondance is given by
(1.7) Λ(X) = ωu0
(X̃), ∀X ∈ g
where X̃ is the vector field on P induced by X (i.e. ∀u∈P, X̃(u)= d
dt |t=0
exp(tX).u)
and Λ: g → k is defined by Λ|m = Λm and Λ|h = λ (hence completely determined by
Λm).
Corollary 1.5.1. In the previous theorem, let us suppose that P is a K-
structure on M = G/H, i.e. P is a subbundle of the bundle L(M) of linear frames
on M with structure group K ⊂ GL(n, R) = GL(m) (we identify as usual m with
Tx0
M by ξ → ξ.x0, and Tx0
M to Rn
via the linear frame u0 ∈ P ⊂ L(M)). Then in
terms of the G-invariant covariant derivative ∇ corresponding to ω, the G-invariant
linear connection in P, the previous bijective correspondance may be given by
Λ(X)(Y ) = ∇X̃Ỹ
where X̃, Ỹ are any (local) left G-invariant vector fields extending X, Y i.e. there
exists a local section, g: U ⊂ M → G, of π: G → M, such that X̃x = Adg(x)(X).p,
for all x ∈ M.
Remark 1.5.1. In theorem 1.5.1, the G-invariant connection in P defined by
Λm = 0 is called the canonical connection (with respect to the decomposition
g = h + m). If we set P(M, K) = G(G/H, H) with group of automorphisms](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-36-320.jpg)
![22 IDRISSE KHEMAR
G, the G-invariant connection defined by the horizontal distribution H(m) is the
canonical connection.
Now, let P be a G-invariant K-structure on M = G/H as in corollary 1.5.1.
Let P
be a G-invariant subbundle of P with structure group K
⊂ K, then the
canonical connection in P
defined by Λm = 0 is (the restriction of ) the canonical
connection in P which is itself the restriction to P of the canonical connection in
L(M). In particular, if we set P
= G.u0, this is a subbundle of P with group H,
which is isomorphic to the bundle G(G/H, H). Then the canonical linear connec-
tion in P
corresponds to the invariant connection in G(G/H, H) defined by the
distribution H(m).
Theorem 1.5.2. Let P ⊂ L(M) be a K-structure on M = G/H. Then the
canonical linear connection (Λm = 0) in P defines the covariant derivative ∇0
in
TM (obtained from H(m) in the associated bundle G×H m ∼
= TM). Moreover there
is a bijective correspondence between the set of of G-invariant linear connections
∇, on M, determined by a connection in P, and the set of linear maps Λm : m →
k ⊂ gl(m) such that
(1.8) Λm(hXh−1
) = Adm(h)Λm(X)Adm(h)−1
∀X ∈ m, ∀h ∈ H,
given by
∇ = ∇0
+ Λ̄m
i.e. ∇X Y = ∇0
XY + Λ̄m(X)Y for any vector fields X, Y on M, where with the help
of ( 1.8) we extended the Ad(H)-equivariant map Λm : m × m → m to the bundle
G ×H m = TM to obtain a map Λ̄m : TM × TM → TM.
Example 1.5.1. Let us suppose that M is Riemannian (i.e. AdmH is compact
and m is endowed with an AdH invariant inner product which defines a G-invariant
metric on M) and let us take P = O(M) the bundle of orthonormal frames on
M, the previous correspondance is between the set of G-invariant metric linear
connections and the set of Ad(H)-equivariant linear maps Λm : m → so(m).
In particular the canonical connection ∇0
is metric (for any G-invariant metric
on M).
Theorem 1.5.3. •: G-invariant tensors on the reductive homogeneous
space M = G/H (or more generally G-invariant sections of associated
bundles) are parallel with respect to the canonical connection.
•: The canonical connection is complete (the geodesics are exactly the curves
xt = exp(tX).x0, for X ∈ m).
•: Let P be a G-invariant K-structure on M = G/H, then the G-invariant
connection defined by Λ: m → k has the same geodesics as the canonical
connection if and only if
Λm(X)X = 0, ∀X ∈ m
Theorem 1.5.4. The torsion tensor T and the curvature tensor R of the G-
invariant connection corresponding to Λm is given at the origin point x0 as follows:
(1) T(X, Y ) = Λm(X)Y − Λm(Y )X − [X, Y ]m,
(2) R(X, Y ) = [Λm(X), Λm(Y )] − Λm([X, Y ]m) − adm([X, Y ]h),
for X, Y ∈ m.
In particular, for the canonical connection we have T(X, Y ) = −[X, Y ]m and
R(X, Y ) = −adm([X, Y ]h), for X, Y ∈ m; moreover we have ∇T = 0, ∇R = 0.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-37-320.jpg)
![1. INVARIANT CONNECTIONS ON REDUCTIVE HOMOGENEOUS SPACES 23
1.6. A family of connections on the reductive space M.
We take in what precedes (i.e. in section 1.5) P = L(M). Then let us consider
the one parameter family of connections ∇t
, 0 ≤ t ≤ 1 defined by
Λt
m(X)Y = t[X, Y ]m, 0 ≤ t ≤ 1.
For t = 0, we obtain the canonical connnection ∇0
. Since for any t ∈ [0, 1],
Λt
m(X)X = 0, ∀X ∈ m, ∇t
has the same geodesics as ∇0
and in particular is
complete. The torsion tensor is given (at x0) by
(1.9) Tt
(X, Y ) = (2t − 1)[X, Y ]m.
In particular ∇
1
2 is the unique torsion free G-invariant linear connection having the
same geodesics as the canonical connection (according to theorems 1.5.3 and 1.5.4).
Assume now that M is Riemannian, and let us take P = O(M). Then ∇t
is
metric if and only if Λt
m takes values in k = so(m) which is equivalent (for t = 0) to
say that M is naturally reductive (which means by definition that ∀X ∈ m, [X, ·]m
is skew symmetric). Now (still in the Riemannian case) let us construct a family
of linear connections,
met
∇t
, 0 ≤ t ≤ 1, which are always metric:
met
∇t
= ∇0
+ t [ , ][m] + UM
where UM
: TM ⊕ TM → TM is the ”natural reductivity term” which is the
symmetric bilinear map defined by1
(1.10) UM
(X, Y ), Z = [Z, X][m], Y + X, [Z, Y ][m]
for all X, Y, Z ∈ [m]. Since UM
is symmetric, the torsion of
met
∇t
is once again given
by
Tt
(X, Y ) = (2t − 1)[X, Y ][m]
and thus
met
∇
1
2 is torsion free and metric and we recover that
met
∇
1
2 is the Levi-Civita
connection
met
∇
1
2 = ∇L.C.
.
Obviously if M is naturally reductive then
met
∇t
= ∇t
, ∀t ∈ [0, 1]. Moreover if M
is (locally) symmetric, i.e. [m, m] ⊂ h, then all these connections coincide and are
equal to the Levi-Civita connection:
met
∇t
= ∇t
= ∇0
= ∇L.C.
.
Remark 1.6.1. ∇1
is interesting since it is nothing but the flat differentiation
in the trivial bundle M × g followed by the projection onto [m] (along [h]) (see
remark 1.4.1). So this connection is very natural and following [1], we will call it
the anticanonical connection.
1UM is the G-invariant extension of Um : m ⊕ m → m, its restriction to m ⊕ m.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-38-320.jpg)
![24 IDRISSE KHEMAR
1.7. Differentiation in End(T(G/H)).
According to section 1.2, we have
End(T(G/H)) = G ×H End(m) ⊂ (G/H) × End(g),
the previous inclusion being given by [g, A] → (π(g), Adg ◦ A ◦ Adg−1
) and we
embed End(m) in g by extending any endomorphism of m to the corresponding
endomorphism of g which vanishes on h. In other words End(T(G/H)) can be iden-
tified to the subbundle [End(m)] of the trivial bundle (G/H) × End(g), with fibers
[End(m)]g.x0
= End(Adg(m)) = Adg(End(m))Adg−1
= Adg(End(m) ⊕ {0})Adg−1
.
Now, let us compute in terms of the Lie algebra setting, the derivative of
the inclusion map I: End(T(G/H)) → M × End(g) or more concretely the flat
derivative in M × End(g) of any section of End(T(G/H)). To do that, we compute
the derivative of
Ĩ: (g, Am) ∈ G × End(m) −→ (g.x0, Adg ◦ Am ◦ Adg−1
) ∈ M × End(g).
We obtain
dĨ(g, Am) = Adg(θm) . π(g), Adg (dAm + [adθ, Am]) Adg−1
.
Next, let us decompose the endomorphisms in g into blocs according to the vector
space decomposition g = h ⊕ m:
(1.11) End(g) =
End(h) End(m, h)
End(h, m) End(m)
.
By regrouping terms, we obtain the following splitting
End(g) = End(m) ⊕ (End(m, h) ⊕ End(h, m) ⊕ End(h)) ,
which applied to dĨ(g, Am), gives us the decomposition
dĨ(g, Am) = 0, Adg (dAm + [admθh, Am] + [[admθm]m, Am]) Adg−1
+ Adg(θm) . π(g), Adg ([admθm]h ◦ Am − Am ◦ adhθm) Adg−1
.
(1.12)
The first term is in the vertical space VĨ(g,Am) =Adg(End(m))Adg−1
=End(Tπ(g)M)
and the previous decomposition (1.12) provides us with a splitting TEnd(M) =
V ⊕ H = π∗
M (End(M)) ⊕ H, i.e. a connection on End(M). Let us determine this
connection: we see that the projection on the vertical space (along the horizontal
space) corresponds to the projection on [End(m)] following (1.11) so that according
to remark 1.6.1, we can conclude that the horizontal distribution H defines the
connection ∇1
on End(TM) = TM∗
⊗ TM.
Remark 1.7.1. a) We can recover this previous fact directly from the first
term of (1.12) and the definition of ∇1
. Indeed, first recall that given two linear
connections ∇, ∇
on M, we can write ∇
= ∇ + F, where F is a section of
TM∗
⊗ End(TM), and then for any section A in End(TM),
∇
A = ∇A + [F, A].
Besides, ∇1
= ∇0
+ [ , ][m], and moreover, if we write (locally) A = (π(U), AdU ◦
Am ◦ AdU−1
) where U is a local section of π and Am ∈ C∞
(M, End(m)), then
according to (1.4),
(1.13) ∇0
A = AdU (dAm + [admαh, Am]) ,](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-39-320.jpg)
![1. INVARIANT CONNECTIONS ON REDUCTIVE HOMOGENEOUS SPACES 25
so that we conclude that
∇1
A = AdU (dAm + [admαh, Am] + [[admαm]m, Am]) AdU−1
which is the (pullback of) the first term of (1.12).
b) Furthermore, if G/H is (locally) symmetric (i.e. [m, m] ⊂ h), then ∇L.C.
=
∇0
= ∇1
and in particular
(1.14) ∇L.C.
A = AdU (dAm + [admαh, Am]) .](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-40-320.jpg)
![CHAPTER 2
m-th elliptic integrable system associated to a
k
-symmetric space
2.0.1. Definition of Gτ
(even when τ does not integrate in G). Here, we will
extend the notion of subgroup fixed by an automorphism of Lie group to the sit-
uation where only a Lie algebra automorphism is provided. Indeed, let τ : G → G
be a Lie group automorphism, then usually one can define Gτ
= {g ∈ G| τ(g) = g}
the subgroup fixed by τ. Now, we want to extend this definition to the situation
where we only have a Lie algebra automorphism, and so that the two definitions
coincide when the Lie algebra automorphism integrates in G.
Let g be a real Lie algebra and τ : g → g be an automorphism. Then let us denote
by
(2.1) g0 = gτ
:= {ξ ∈ g| τ(ξ) = ξ}
the subalgebra of g fixed by τ. Let us assume that τ defines in g a τ-invariant
reductive decomposition
g = g0 ⊕ n, [g0, n] ⊂ n, τ(n) = n.
Moreover we suppose that we have n = Im (Id − τ), i.e. that the decomposition
g = g0 ⊕ n coincides with the Fitting decomposition of Id − τ (remark that this
decomposition is then automatically reductive since adξ ◦ (Id − τ) = (Id − τ) ◦ adξ,
∀ξ ∈ g0).
Without loss of generality, we assume that the center of g is trivial. Moreover,
we assume also, without loss of generality, that g0 does not contain a nontrivial
ideal of g, i.e. that adn : g0 → gl(n) is injective (the kernel is a τ-invariant ideal of
g that we factor out). We then have
(2.2) g0 = {ξ ∈ g| τ ◦ adξ ◦ τ−1
= adξ}
Let G be a connected Lie group with Lie algebra g. Then let us consider the
subgroup
G0 = {g ∈ G| τ ◦ Adg ◦ τ−1
= Adg}.
Then G0 is a closed subgroup of G and Lie G0 = g0.
Moreover, without loss of generality, we will suppose that G0 does not contain
non-trivial normal subgroup of G (by factoring out, if needed, by some discrete
subgroup of G), i.e. that Adn : G0 → GL(n) is injective (see section 1). Now,
we want to prove that if τ integrates in G, then we have G0 = Gτ
, where Gτ
is the subgroup fixed by τ : G → G. First, we have ∀g ∈ Gτ
, τ ◦ Adg ◦ τ−1
=
Adτ(g) = Adg, thus Gτ
⊂ G0. Conversely, we have ∀g ∈ G0, Adg(n) = n and
Adg = τ ◦ Adg ◦ τ−1
= Adτ(g) so that Adng = Adnτ(g) and thus g = τ(g) since
Adn : G0 → GL(n) is injective. We have proved Gτ
= G0. This allows us to make
the following:
27](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-42-320.jpg)
![28 IDRISSE KHEMAR
Definition 2.0.1. Let g be a real Lie algebra and τ : g → g be an automor-
phism, and G a connected Lie group with Lie algebra g. Let us assume that τ defines
in g a τ-invariant reductive decomposition: g = g0 ⊕ n with n = Im (Id − τ). Then
we will set
Gτ
:= {g ∈ G| τ ◦ Adg ◦ τ−1
= Adg}.
Let us conclude this subsection by some notation:
Notation and conventions. In all the paper, when a real Lie algebra g and an
automorphism τ will be given, then we will suppose without loss of generality that
the center of g is trivial, g0 will denote the Lie subalgebra defined by (2.1), G will
denote a connected Lie group with Lie algebra g and G0 ⊂ G a closed subgroup
such that (Gτ
)0
⊂ G0 ⊂ Gτ
(which implies that its Lie algebra is g0).
Moreover, without loss of generality, we will always suppose that g0 does not
contain non-trivial ideal of g - we will then say that (g, g0) is effective - and also
suppose that Gτ
does not contain non-trivial normal subgroup of G (by factoring
out, if needed, by some discrete subgroup of G). Consequently, when τ can be
integrated in G, then Gτ
will coincide with the subgroup of G fixed by τ : G → G.
Remark 2.0.2. • Let g be a Lie algebra and τ : g → g be an automorphism.
Let us consider G
= AdG =: Int(g) the adjoint group of g (which does not depend
on the choice of the connected group G), and let C = ker Ad be the center of G.
Then we can identify the morphism Ad with the covering π: G → G/C and G
to
G/C. Besides τ always integrates in G
into τ
defined by τ
= Intτ : Adg ∈ G
→
τ ◦ Adg ◦ τ−1
and we have τ
◦ π = π ◦ τ (See [36, p. 127]). Then, we see that
G0 = Ad−1
(Gτ
) = π−1
(Gτ
).
• Now, let us come back to the hypothesis of trivialness of the center of g. Let
g be a real Lie algebra with center c and τ : g → g an automorphism. Then we
have τ(c) = c. Therefore, if τ is of order k
, for any m ∈ N∗
, the m-th elliptic
integrable system associated to τ splits into two independent systems: a general
one on g
= g/c and a trivial one on c = Rn
: dαλ = 0, ∀λ ∈ C∗
. Then the solutions
of this trivial system are the maps fλ : L → c, fλ =
m
j=−m λj
fj, ∀λ ∈ C∗
, such
that f−j = fj, j ≥ 0, and f−j is an holomorphic map into the eigenspace cC
∩ gC
−j.
(Of course αλ = dfλ.) Therefore, the hypothesis of trivialness of the center can be
done without loss of generality.
2.1. Finite order Lie algebra automorphisms.
Let g be a real Lie algebra and τ : g → g be an automorphism of order k
.
Let ωk be a k
-th primitive root of unity. Then we have the following eigenspace
decomposition:
gC
=
j∈Z/kZ
gC
j , [gC
j , gC
l ] ⊂ gC
j+l
where gC
j is the ωj
k -eigenspace of τ.
We then have to distinguish two cases.
2.1.1. The even case: k
= 2k. Then we have gC
0 = (g0)C
. Moreover let us
remark that
(2.3) gC
j = gC
−j, ∀j ∈ Z/k
Z.
Therefore gC
k = gC
−k = gC
k so that we can set gC
k = (gk)C
with
gk = {ξ ∈ g| τ(ξ) = −ξ}.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-43-320.jpg)
![m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k
-SYMMETRIC SPACE 29
Moreover, owing to (2.3), we can define mj as the unique real subspace in g such
that its complexified is given by
mC
j = gC
j ⊕ gC
−j for j = 0, k,
and n as the unique real subspace such that
nC
=
j∈Z
k{0}
gC
j ,
that is n = (⊕k−1
j=1 mj) ⊕ gk. In particular τ defines a τ-invariant reductive decom-
position g = g0 ⊕ n.
Hence the eigenspace decomposition is written:
gC
=
gC
−(k−1) ⊕ . . . ⊕ gC
−1
⊕ gC
0 ⊕ gC
1 ⊕ . . . ⊕ gC
k−1 ⊕ gC
k
so that by grouping
gC
= gC
0 ⊕ gC
k ⊕
⊕k−1
j=1 mC
j
= hC
⊕ mC
where h = g0 ⊕ gk and m = ⊕k−1
j=1 mj. Considering the automorphism σ = τ2
, we
have h = gσ
and g = h ⊕ m is the reductive decomposition defined by the order
k automorphism σ. Without loss of generality, and according to our convention
applied to g and σ, we will suppose in the following that (g, h) is effective i.e. h
does not contain a nontrivial ideal of g. This implies in particular that (g, g0) is
also effective.
Now let us integrate our setting: let G be a Lie group with Lie algebra g
and we choose G0 such that (Gτ
)0
⊂ G0 ⊂ Gτ
. Then G/G0 is a (locally) 2k-
symmetric space (it is globally 2k-symmetric if τ integrates in G) and is in particular
a reductive homogeneous space (reductive decomposition g = g0 ⊕ n).
Moreover since σ = τ2
is an order k automorphism, then for any subgroup
H, such that (Gσ
)0
⊂ H ⊂ Gσ
, G/H is a (locally) k-symmetric space. In all
the following we will always do this choice for H and suppose that H ⊃ G0 (it is
already true up to covering since h ⊃ g0) so that N = G/G0 has a structure of
associated bundle over M = G/H with fibre H/G0: G/G0
∼
= G ×H H/G0. We can
add that on h, τ is an involution: (τ|h)2
= Idh, whose symmetric decomposition is
h = g0 ⊕ gk, and gives rise to the (locally) symmetric space H/G0. The fibre H/G0
is thus (locally) symmetric (and globally symmetric if the inner automorphism
Intτ|m stabilizes AdmH). Owing to the effectivity of (g, h), we have the following
characterization:
g0 = {ξ ∈ h|[admξ, τ|m] = 0}
(2.4)
gk = {ξ ∈ h|{admξ, τ|m} = 0}
(2.5)
{} being the anticommutator.
Two different types of 2k-symmetric spaces. Since (τ|m)k
is an involution,
there exist two invariant subspaces m
and m
, of m, each sum of certain mj’s, such
that
(τ|m)k
= −Idm ⊕ Idm .](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-44-320.jpg)
![30 IDRISSE KHEMAR
These subspaces m
and m
can be computed easily : m
= ⊕
[ k−2
2 ]
j=0 m2j+1 and m
=
⊕
[ k−1
2 ]
j=1 m2j. In other words, their complexifications are given by
mC
=
zk
= −1
z = −1
ker(τ − zId), mC
=
zk
= 1
z = ±1
ker(τ − zId).
At this stage, there are two possibilities:
•: if m
= 0 then (τ|m)k
= −Idm and τ|m admits eigenvalues only in the set
{zk
= −1, z = −1}.
•: if m
= 0 then (τ|m)k
= −Idm and τ|m admits eigenvalues in each of the
sets {zk
= 1, z = ±1} and {zk
= −1, z = −1}.
These two cases give rise to two different types of 2k-symmetric spaces (see sec-
tion 3.5)
G-invariant metrics. Now, let us suppose that M = G/H is Riemannian (i.e.
AdmH is compact) then we can choose an AdH-invariant inner product on m for
which τ|m is an isometry1
. In the next of the paper, we will always choose this kind
of inner product on m. Therefore, τ|m is an order 2k isometry. We will study this
kind of endomorphism in section 3.
Moreover, let us remark that if G/H is Riemannian then so is G/G0. Further,
since the elliptic system we will study in this paper is given in the Lie algebra setting
it is useful to know how the fact that G/H is Riemannian can be read in the Lie
algebra setting. In fact, under our hypothesis of effectivity, G/H is Riemannian
if and only if h is compactly imbedded2
in g and AdH/AdH0
is finite. Moreover,
according to proposition 8.2.3 in the Appendix, AdH/AdH0
is always finite so that
G/H is Riemannian if and only if h is compactly imbedded in g.
2.1.2. The odd case: k
= 2k + 1. As in the even case we have gC
0 = (g0)C
and
gC
j = gC
−j, ∀j ∈ Z/k
Z. Then we obtain the following eigenspace decomposition:
(2.6) gC
= gC
−k ⊕ . . . ⊕ gC
−1 ⊕ gC
0 ⊕ gC
1 ⊕ . . . ⊕ gC
k ,
which provides in particular the following reductive decomposition:
g = g0 ⊕ m
with m = ⊕k
j=1mj and mj is the real subspace whose the complexification is mC
j =
gC
−j ⊕ gC
j . According to our convention, we suppose that (g, g0) is effective.
Then, as in the even case, integrating our setting and choosing G0 such that
(Gτ
)0
⊂ G0 ⊂ Gτ
, we consider N = G/G0 which is a locally (2k + 1)-symmetric
space and in particular a reductive homogeneous space. Moreover, the decomposi-
tion (2.6) gives rises to a splitting TNC
= T1,0
N ⊕ T0,1
N defined by
(2.7)
TNC
= ⊕k
j=1[gC
−j] ⊕ ⊕k
j=1[gC
j ]
= T1,0
N ⊕ T0,1
N
This splitting defines a canonical almost complex structure on G/G0, that we will
denote by J.
1See the Appendix, theorem 8.2.2, for the proof of the existence of such an inner product.
2See [36, p. 130] for a definition.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-45-320.jpg)
![m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k
-SYMMETRIC SPACE 31
Let us suppose that N = G/G0 is Riemannian; then the subgroup generated by
AdmG0 and τ|m is compact (because AdmG0 is compact and τ|mAdmg τ−1
|m = Admg,
∀g ∈ G0, and τm is of finite order). Therefore there exists an AdG0-invariant inner
product on m for which τ|m is an isometry. In the next of the paper, we will always
choose this kind of inner product on m (when N is Riemannian).
2.2. Definitions and general properties of the m-th elliptic system.
2.2.1. Definitions.
Let τ : g → g be an order k
automorphism with k
∈ N∗
(if k
= 1 then τ = Id).
We use the notations of 2.1. Let us begin by defining some useful notations.
Notation and convention Given I ⊂ N, we denote by
j∈I gC
j , the product
j∈I gC
j mod k . In the case
j∈I gC
j mod k is a direct sum in gC
, we will identify it
with the previous product via the canonical isomorphism
(2.8) (aj)j∈I −→
j∈I
aj,
and we will denote these two subspaces by the same notation ⊕j∈I gC
j .
Now, let us define the m-th elliptic integrable system associated to a k
-
symmetric space, in the sense of Terng [66].
Definition 2.2.1. Let L be a Riemann surface. The m-th (g, τ)-system (with
the (−)-convention) on L is the equation for (u0, . . . , um), (1, 0)-type 1-form on L
with values in
m
j=0 gC
−j:
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
¯
∂uj +
m−j
i=0
[ūi ∧ ui+j] = 0 (Sj), if 1 ≤ j ≤ m,
¯
∂u0 + ∂ū0 +
m
j=0
[uj ∧ ūj] = 0 (S0)
(Syst)
It is equivalent to say that the 1-form
(2.9) αλ =
m
j=0
λ−j
uj + λj
ūj =
m
j=−m
λj
α̂j
satisfies the zero curvature equation:
(2.10) dαλ +
1
2
[αλ ∧ αλ] = 0, ∀λ ∈ C∗
.
Definition 2.2.2. Let L be a Riemann surface. The m-th (g, τ)-system (with
the (+)-convention) on L is the equation (Syst) as in definition 2.2.1 but for
(u0, . . . , um), (1, 0)-type 1-form on L with values in
m
j=0 gC
j :3
It is equivalent to say that the 1-form
(2.11) αλ =
m
j=0
λj
uj + λ−j
ūj =
m
j=−m
λj
α̂j
satisfies the zero curvature equation ( 2.10).
3instead of
m
j=0 gC
−j.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-46-320.jpg)
![32 IDRISSE KHEMAR
Remark 2.2.1. The difference between the two conventions is that in the first
one α
λ =
m
j=0 λ−j
uj involves negative powers of λ whereas in the second one α
λ
involves positive powers of λ (in other words α̂
−j = 0, for j ≥ 1 in the first one
whereas α̂
j = 0, for j ≥ 1 in the second one). In fact the second system is the first
system associated to τ−1
and vice versa.
The first convention is the traditional one: it was used for harmonic maps into
symmetric space (see [23]) and by Hélein-Romon [33, 34, 35] for Hamiltonian
stationary Lagrangian surfaces in Hermitian symmetric space – first example of
second elliptic integrable system associated to a 4-symmetric space. Then this
convention was used in [42, 43, 44]. Terng [66], herself, in her definition of the
elliptic integrable system uses also this convention. However in [17], it is the second
convention which is used.
The (+)-convention is in fact the most natural, as we will see, since in the (−)-
convention, there is a minus sign which appears when we pass from the Lie algebra
setting to the geometric setting. This is what happens for example when we will
associate to the automorphism τ, a almost complex complex structure in the target
space (see also [44], remark 13). Namely, we will consider that the eigenspace
spaces g−j will define the (1, 0)-part and the subspaces gj the (0, 1)-part of the
tangent space of the target. But the (+)-convention leads to several changes in the
traditional conventions, like for example in the DPW method [23], we must use the
Iwasawa decomposition ΛGC
τ = ΛGτ .Λ−
B GC
τ instead of ΛGC
τ = ΛGτ .Λ+
B GC
τ and in
particular the holomorphic potential involves positive power of λ instead of negative
one as it is the case traditionally. We decided here to continue to perpetuate the
tradition as in [44] and to use the first convention. So in the following when we
will speak about the m-th elliptic integrable system, it will be according
to the definition 2.2.1.
Notation. Sometimes, to avoid confusion we will denote (Syst) either by
(Syst(m, g, τ)), (Syst(m, τ)) or simply by (Syst(m)) depending on the context and
the needs.
For shortness we will also often say the (m, g, τ)-system instead of the m-th
(g, τ)-system. We will also say the m-th elliptic (integrable) system associated to
(the k
-symmetric space) G/G0.
We will say that a family of 1-forms (αλ)λ∈C∗ (denoted by abuse of notation,
simply by αλ) is solution of the (m, g, τ)-system (or of (Syst)) if it corresponds
to some solution u of this system, according to (2.9). Therefore αλ is solution of
the (m, g, τ)-system if and only if it can be written in the form (2.9), for some
(1, 0)-type 1-form u on L with values in
m
j=0 gC
−j, and satisfies the zero curvature
equation (2.10).
It will turn out that the (m, g, τ)-system has distinctively different behaviour
if k
is even and if k
is odd. Moreover, for every k
there are three different types
of behaviour, according to the size of m relatively to k
.
Definition 2.2.3. If k
= 1, set mk = 0. If k
1, we set mk =
k
+ 1
2
=
k if k
= 2k
k + 1 if k
= 2k + 1
.
We will say that the m-th (g, τ)-system is:](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-47-320.jpg)
![m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k
-SYMMETRIC SPACE 33
– in the primitive case (or that the system is primitive) if 0 ≤ m mk ,
– in the determined case (or that the system is determined) if mk ≤ m ≤ k
− 1,
– and in the underdetermined case (or that the system is underdetermined) if
m k
− 1.
Moreover, the determined system of minimal order mk will be called ”the min-
imal determined system”, and the one of maximal order k
− 1 will be called ”the
maximal determined system”.
Let us consider the g-valued 1-form α := αλ=1. Then we have α =
m
j=0 uj +ūj
according to (2.9) which is equivalent to α
=
m
j=0 uj, since α is g-valued
• In the primitive and determined cases (m ≤ k
− 1),
m
j=0 gC
−j is a direct
sum so that u = (u0, . . . , um) can be identified with
m
j=0 uj = α
via (2.8) and
according to our convention. We will then write simply u = α
. In particular we
have
uj = α
−j ∀j, 0 ≤ j ≤ m
with αj := [α]gj
∀j ∈ Z/k
Z. Hence in the primitive and determined cases the
m-th (g, τ)-system can be considered as a system on α. Moreover, we can recover
αλ from α and we will speak about the ”extended Maurer Cartan form” αλ which
is then associated to α by
αλ =
m
j=1
λ−j
α
−j + α0 +
m
j=1
λj
α
j
according to (2.9).
• In the underdetermined case,
m
j=0 gC
−j is not a direct sum so that to a given
α (coming from some solution αλ of the m-th (g, τ)-system, according to α = αλ=1)
there are a priori many (other) corresponding solutions u = (u0, . . . , um) since
∀j ∈ Z/k
Z, α
−j =
i≡j[k]
ui.
In fact, we will prove that there are effectively an infinity of other solutions αλ
satisfying the condition αλ=1 = α (see theorem 2.2.1; see also section 2.5 for a
conceptual explanation).
2.2.2. The geometric solution. Convention. Our study, in the present paper,
is local therefore we will suppose (when it is necessary to do so) either that L is
simply connected or that all lifts (of maps defined on L) and integrations (of 1-
forms on L) are made locally. We consider that these considerations are implicit
and will not specify these most of the time.
The equations (2.10) and (2.9) are invariant by gauge transformations by the
group C∞
(L, G0):
U0 · αλ = AdU0(αλ) − dU0.U−1
0 .
where U0 ∈ C∞
(L, G0). This means that if αλ satisfies (2.10) then so is U0 · αλ
and if αλ can be written in the form (2.9) then so is U0 · αλ. Therefore if αλ is a
solution of (Syst) then so is U0 ·αλ. This allows us to define a geometric solution
of (Syst) as follows:
Definition 2.2.4. A map f : L → G/G0 is a geometric solution of (Syst)
if for any (local) lift U of f, into G, there exists a (local) solution αλ of (Syst)
such that U−1
.dU = αλ=1.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-48-320.jpg)
![34 IDRISSE KHEMAR
In other words, we obtain the set of geometric solutions as follows: for each
solution αλ of (Syst), consider the g-valued 1-form α := αλ=1, then integrate it
by U : L → G, U−1
.dU = α, and finally project U on G/G0 to obtain the map
f : L → G/G0.
Now, to simplify the exposition, let us suppose that L is simply connected
(until the end of 2.2.2). Then (αλ)λ∈C∗ → α = αλ=1 is a surjective map from the
set of solutions of (Syst) to the set of Maurer-Cartan forms of lifts of geometric
solutions. According to the discussion at the end of subsection 2.2.1, this map is
bijective in the primitive and determined case (m ≤ k
− 1) and not injective in the
underdetermined case (m k
− 1). By quotienting by C∞
(L, G0), we obtain a
surjective map πm with the same properties, taking values in the set of geometric
solutions
Let us make more precise all that. We suppose, until the end of this sub-
section 2.2.2, that the automorphism τ : g → g is fixed (so that the only data
which varies in the (m, g, τ)-system is the order m). First, let us give an explicit
expression of the space S(m) of solutions αλ of the system (Syst(m)), i.e. the so-
lutions of the zero curvature equation (2.10), which satisfies the equality (2.9) for
some (1, 0)-type 1-form u on L with values in
m
j=0 gC
−j. To do that, we want to
express the condition to be written in the form (2.9) as a condition on the loop
α• : λ ∈ S1
→ αλ ∈ C(T∗
L⊗g). The ” • ” means of course functions on the param-
eter4
λ ∈ S1
. More precisely, we will consider α• as a 1-form on L with values in
the loop Lie algebra C(S1
, g). Then the condition to be written in the form (2.9)
means:
(2.12) (2.9) ⇐⇒ (α• ∈ Λmgτ and α
• ∈ Λ−
gC
τ )
where
Λgτ = {η• : S1
→ g| ηωλ = τ(ηλ), ∀λ ∈ S1
}
Λmgτ = {η• ∈ Λgτ | ηλ =
|j|≤m
λj
η̂j}
Λ−
gC
τ = {η• ∈ ΛgC
τ | ηλ =
j≤0
λj
η̂j}
and ω is a k
-th primitive root of unity. We refer the reader to [60] for more details
about loop groups ad their Lie algebras (in particular about the possible choices
of topology which makes Λgτ be a Banach Lie algebra). Therefore the space of
solutions of (Syst(m)) is given by
(2.13) S(m) = {α• ∈ C(T∗
L ⊗ Λmgτ )| α
• ∈ Λ−
gC
τ and dα• +
1
2
[α• ∧ α•] = 0}.
Let us remark that the condition α
• ∈ Λ−
gC
τ can be interpretated as a condition of
C-linearity. Indeed, the Banach vector space Λgτ /g0 is naturally endowed with the
complex structure defined by the following decomposition
(2.14) (Λgτ /g0)C
= ΛgC
τ /gC
0 = Λ−
∗ gτ ⊕ Λ+
∗ gτ ,
where Λ±
∗ gτ = {η• ∈ ΛgC
τ | ηλ =
j≷0 λj
η̂j}. Then the condition α
• ∈ Λ−
gC
τ
means that [α
•]∗ : TL → (Λgτ /g0)C
is C-linear, where [ ]∗ denotes the component
in Λ∗gτ = {η• ∈ Λgτ |ηλ =
j=0 λj
η̂j} ∼
= Λgτ /g0.
4Remark that α• determines (αλ)λ∈C∗ , when this latter satisfies (2.9).](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-49-320.jpg)
![m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k
-SYMMETRIC SPACE 35
Now let us integrate our setting. Firstly, let us define the twisted loop group
([60])
ΛGτ = {U• : S1
→ G|Uωλ = τ (Uλ)}.
Then, let us set
Em
= {U• : L → ΛGτ |Uλ(0) = 1, ∀λ ∈ S1
; αλ := U−1
λ .dUλ is a solution of
(Syst(m))}
Em
1 = {U : L → G|∃U• ∈ Em
, U = U1}
Gm
1 = {f : L → G/G0 geometric solution of (Syst(m)), f(0) = 1.G0}
Gm
= {f• = πG/G0
◦ U•, U• ∈ Em
}
Remark that because of the gauge invariance: E(m).K ⊂ E(m), where K
= C∞
∗ (L, G0) = {U ∈ C∞
(L, G0)|U(0) = 1}, any lift5
U• : L → ΛGτ of an ele-
ment f• ∈ Gm
belongs to Em
.
Definition 2.2.5. An element f• ∈ Gm
will be called an extended geometric
solution of (Syst(m)).
The space of geometric solutions is obviously obtained from the space of ex-
tended geometric solutions Gm
by the evaluation at λ = 1. Moreover S(m) Em
is determined by Gm
• because of the gauge invariance: E(m).K ⊂ E(m), so that we
can write Gm
= E(m)/K. Consequently, we have also Gm
1 = Em
1 /K.
Finally, we obtain the following diagram
S(m)
int
−
−
−
−
→
∼
=
Em πK
−
−
−
−
→ Em
/K Gm
•
ev1
⏐
⏐
⏐
⏐
⏐
⏐
πm
⏐
⏐
S(m)1
int
−
−
−
−
→
∼
=
Em
1 −
−
−
−
→ Em
1 /K Gm
.
Therefore, the surjective map πm is bijective for m ≤ k
− 1 and not injective for
m k
− 1 (because so is ev1). We will need the following definition:
Definition 2.2.6. Given a g-valued Maurer-Cartan 1-form α on L, we define
the geometric map corresponding to α, as f = πG/G0
◦ U, where U integrates
α: U−1
.dU = α, U(0) = 1.
We have seen that in the primitive and determined cases, we can consider
(Syst(m)) as a system on the g-valued 1-form α := αλ=1. Since the Maurer-Cartan
equation for α is always contained in (Syst(m)) according to (2.10), this systems
on α is itself equivalent to a system on the geometric map f corresponding to α.
This system on f is then a G-invariant elliptic PDE on f of order ≤ 2.
Let us summarize:
Proposition 2.2.1. The natural surjective map πm : Gm
→ Gm
1 from the set
of extended geometric solutions of the (m, g, τ)-system into the set of geometric
solutions is bijective in the primitive and determined cases (m ≤ k
− 1) and not
injective in the underdetermined case (m k
− 1). Moreover, in the primitive and
determined cases, the (m, g, τ)-system - which is initially a system on the Λmgτ -
valued 1-form αλ - is in fact a system on the 1-form α := αλ=1, itself equivalent to
an elliptic PDE of order ≤ 2 on the corresponding geometric map f : L → G/G0.
5With initial condition U(0) = 1.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-50-320.jpg)
![36 IDRISSE KHEMAR
Furthermore, let us interpret the C-linearity of [α
•]∗ : TL → (Λgτ /g0)C
in
terms of the corresponding extended geometric solution f• : L → ΛGτ /G0, defined
by f• = πG/G0
◦ U• where U• integrates α•. Firstly, the complex structure defined
in Λgτ /g0 by (2.14) is AdG0-invariant so that it defines a ΛGτ -invariant complex
structure on the homogeneous space ΛGτ /G0. Therefore the C-linearity of [α
•]∗
means exactly that f• : L → ΛGτ /G0 is holomorphic. Now, let us interpret the
condition α• ∈ Λmgτ in terms of the map f•. Let us consider the following AdG0-
invariant decomposition
Λgτ /g0 = Λm∗gτ ⊕ Λmgτ
where Λm∗gτ = Λmgτ ∩ Λ∗gτ and Λmgτ = {η• ∈ Λgτ | ηλ =
|j|m λj
η̂j}, which
gives rise respectively to some ΛGτ -invariant splitting
T(ΛGτ /G0) = HΛ
m ⊕ VΛ
m.
Then HΛ
m and VΛ
m inherit respectively the qualificatifs horizontal and vertical sub-
bundle respectively. Therefore, in the same spirit as [23] (remark 2.5 and proposi-
tion 2.6), the equation (2.13) gives us the following familiar twistorial characteri-
zation
Proposition 2.2.2. A map f• : L → ΛGτ /G0 is an extended geometric solution
of the (m, g, τ)-system if and only if it is holomorphic and horizontal.
2.2.3. The increasing sequence of spaces of solutions: (S(m))m∈N. Again, we
suppose in all 2.2.3 that the automorphism τ is fixed and that L is simply connected.
Then according to the realisation of (Syst(m)) in the forms (2.10) and (2.9), we see
that any solution of (Syst(m)) is solution of (Syst(m
)) for m ≤ m
(take uj = 0
for m j ≤ m
). More precisely, (Syst(m)) is a reduction of (Syst(m
)): (Syst(m))
is obtained from (Syst(m
)) by putting uj = 0, m j ≤ m
, in (Syst(m
)). In
particular, S(m) ⊂ S(m
) for m ≤ m
; so that any solution in the primitive case
(m mk ) is solution of any determined system (mk ≤ m ≤ k
− 1), and any
solution of a determined system is solution of any underdetermined system (m
k
− 1).
{Primitive case} ⊂ {determined case} ⊂ {underdetermined case}.
Remark 2.2.2. We have πm |Gm = πm if m ≤ m
. In particular, πm (Gm
) =
Gm
1 . We can set S(∞) = ∪m∈NS(m), E∞
= ∪m∈NEm
and G∞
= ∪m∈NGm
. Then
we have G∞
= E∞
/K. Moreover we can define the surjective map π∞ : G∞
→ G∞
1
such that π∞|Gm = πm, ∀m ∈ N. Then π∞|Gm is a bijection onto Gm
1 for each
m ≤ k
− 1.
We can call S(∞) the (g, τ)-system, and then we can speak about its subsystem
of order m, namely S(m). In particular, we have the following characterization:
S(∞) = {α• ∈ C(T∗
L ⊗ Λ(∞)gτ )| α
• ∈ Λ−
gτ and dα• +
1
2
[α• ∧ α•] = 0}
where Λ(∞)gτ = ∪m∈NΛmgτ .
Important remark. It could happens that the eigenspaces gj vanishes for
the first values of j, i.e. j close to 0. For example it is a priori possible that g0 = 0
(which implies that our k
-symmetric space is a group). Then for the values of
m ≥ k
close to k
, the underdetermined systems Syst(m) coincide trivially with the
determined system Syst(k
−1), because then uj ∈ g−j = {0}, for k
≤ j ≤ m. This](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-51-320.jpg)
![m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k
-SYMMETRIC SPACE 39
The minimal determined case splits into two cases.
2.3.1. The even minimal determined case: k
= 2k and m = k. Let us recall
the following decomposition
gC
=
gC
−(k−1) ⊕ . . . ⊕ gC
−1
⊕ gC
0 ⊕ gC
1 ⊕ . . . ⊕ gC
k−1 ⊕ gC
k .
It is useful for the following to keep in mind that k = −k mod 2k.
Proposition 2.3.1. The system (Syst(k, τ)) can be written
(2.16)
⎧
⎪
⎨
⎪
⎩
α
−j = 0, 1 ≤ j ≤ k − 1 (Hj)
dα +
1
2
[α ∧ α] = 0 (MC)
¯
∂α
−k + [α
0 ∧ α
−k] = 0 (Sk)
.
More precisely the equations (Sj), 0 ≤ j ≤ k − 1, of (Syst(k, τ)) are respectively the
projection on gC
−j, 0 ≤ j ≤ k − 1, of (MC) (owing to the holomorphicity conditions
(Hj) given by proposition 2.2.3).
Proof. The equation (Sj), for 1 ≤ j ≤ k − 1, is written in terms of α:
¯
∂α
−j +
k−j
i=0
[α
i ∧ α
−j−i] = 0,
according to definition 2.2.1. Since we have α
−j = 0, 1 ≤ j ≤ k − 1, according to
proposition 2.2.3, this equation is equivalent to
(2.17) dα−j +
k−j
i=0
[αi ∧ α−j−i] = 0
which is nothing but the projection of (MC) on gC
−j. Indeed, we have
1
2
[α∧α]gC
−j
=
1
2
⎛
⎝
i+l=−j
[α
i ∧ α
l]+
i+l=−j
[α
i ∧ α
l ]
⎞
⎠=
i+l=−j
[α
i ∧α
l]=
k
i=−k+1
[α
i ∧α
−j−i]
=
k−j
i=0
[α
i ∧ α
−j−i]
where we have used, in the last line, the holomorphicity conditions: α
i = 0 if
−k + 1 ≤ i ≤ −1 and α
−j−i = 0 if k − j ≤ i ≤ k.
Now, since α is real (i.e. is g-valued), this equation (2.17) is also equivalent to
its complex conjugate i.e. the projection of (MC) on gC
j . Moreover, the equation
(S0) is written in terms of α:
¯
∂α
0 + ∂α
0 +
k
j=0
[α
−j ∧ α
j ] = 0
and moreover, using the holomorphicity conditions, we have
k
j=0
[α
−j ∧ α
j ] =
k−1
j=0
[α−j ∧ αj] +
1
2
[α−k ∧ αk] =
1
2
[α ∧ α]g0
which proves that (S0) is equivalent to the projection of (MC) on g0. Finally, the
equation (Sk) is written in terms of α as in (2.16). This completes the proof.](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-54-320.jpg)
![3
Πρώτοι ας έλθουνε οι Σουλιώταις,
Και απ' το Λείψανον αυτό
Ας μακραίνουνε οι προδόταις,
Και απ' τα λόγια οπού θα πω.
4
Φλάμπουρα, όπλα τιμημένα,
Ας γυρθούν κατά τη γη,
Καθώς ήτανε γυρμένα
Εις του Μάρκου τη θανή,
5
Που 'βαστούσε το μαχαίρι,
Όταν του λειψε η ζωή,
Μέσ' 'ς το ανδρόφονο το χέρι,
Και δεν τ' άφινε να βγη. [56]
6
Αναθράφηκε ο γενναίος
'Σ των αρμάτων την κλαγγή·
Τούτον έμπνευσε, όντας νέος,
Μία θεά μελωδική. —
7
Με ταις θείαις της αδελφάδες
Εστεκότουν σιωπηλή,
Ενώ αυξαίνανε οι λαμπράδαις
'Σ του Θεού την κεφαλή,
8
Που εμελέτουνε τη Χτίση·
Και ότι εβγήκε η προσταγή,
Οπού εστένεψε τη Φύση
Αιφνιδίως να φωτιστή,
9](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-57-320.jpg)
![Ασυντρόφιαστη, αν ξανοίξη
Ταις περίστασαις δειναίς,
22
Κι' αν ταις εύρη ευτυχισμέναις,
Νάλθη αντίς για τον εχθρό,
Μ' άλλαις άλυσαις φτειασμέναις
Αποκάτου απ' το Σταυρό,
23 [57]
Πούχε λάβη 'ς ταις αγκάλαις
Από μας, κ' είχε θεούς
Αστραπαίς, ανεμοζάλαις,
Και βρονταίς και ποταμούς.
24
Μόνον τ' αδικοσφαμένα
Τα παιδιά σου, στρυμωχτά,
Με τα χέρια τσακισμένα
Σε εσπρώξανε ομπροστά,
25
Και Συ εχύθηκες, πετώντας
Μία ματιά 'ς τον Ουρανό,
Που τα δίκια σου θωρώντας,
Απεκρίθηκε· Είμαι δω.
26 [58]
Και χτυπώντας ξεθυμαίνει
Εις το πέλαγο, εις τη γη,
Η ρομφαία σου πυρωμένη
Οχ την Άπλαστη Φωνή,
27
Και θαυμάσια τόσα πράχτει,
Οπού οι Τύραννοι της γης](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-60-320.jpg)
![Από βάρβαρο ποδάρι,
Πάρεξ όταν χαλαστή.
52
Δεν ήταν τη μέρα τούτη
Μοσχολίβανα, ψαλμοί.
Νά, μολίβια, νά, μπαρούτι,
Νά, σπαθιών λαμποκοπή.
53
'Σ τον αέρα ανακατώνονται
Οι σπιθόβολοι καπνοί,
Και από πάνου φανερώνονται
Ήσκιοι θείοι πολεμικοί·
54
Και είναι αυτοί, που πολεμώντας
Εσκεπάσανε τη γη,
Πάνου εις τ' άρματα βροντώντας
Με το ελεύθερο κορμί
55 [59]
Και αγκαλιάσματα εκεί πλήθια,
Δάφναις έλαβαν, φιλιά,
Όσα ελάβανε εις τα στήθια
Βόλια τούρκικα σπαθιά.
56
Όλοι εκείνοι οι πολεμάρχοι
Περιζώνουνε πυκνοί
Την ψυχή του Πατριάρχη,
Που τον πόλεμο ευλογεί·
57
Και αναδεύονται, και γέρνουν,
Και εις το πρόσωπο ιλαροί,](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-65-320.jpg)
![Ω! τι κάνετε;
Πού πάτε; Για φερθήτε ειρηνικά·
82
»Γιατί αλλιώς θε να βρεθήτε
»Ή με ξένο βασιλιά,
»Ή θα καταφανισθήτε
»Από χέρια αγαρηνά.»(10)
83
Αφού εδώ 'ς την παλαιά σου
Κατοικία και άλλη φορά
Με διχόνοιαις τα παιδιά σου
Σου ετοιμάσανε εξοριά,
84
Από τότες οπού εσώθη
'Σ την Ελλάδα ο Στρατηγός,
Οπού ο Έλληνας ειπώθη
(Και τώρα όχι) ο στερινός,
85
Έως που ο κόσμος εβαστούσε
Τον απάνθρωπον Αλή,
Που όσον αίμα και αν ρουφούσε
Τόσο εγύρευε να πιή,
86 [60]
Επερνούσαν οι αιώνες
Ή σε ξένη υποταγή,
Ή με ψεύτικαις κορώναις,
Ή με σίδερα και οργή·
87
Και ήλθε τότες και επερπάτει (11)
Όπου επάταγες Εσύ,](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-70-320.jpg)
![Ύπνος έξαφνα σ' επήρε,
Που δεν έχει ξυπνημό·
136
Είναι αδιάφορο, δε βλάβει,
Αν εκεί σιμοτινό
Πλέξη ή τούρκικο καράβι,
Ή καράβι ελληνικό.
137
Άκου, Μπάιρον, πόσον θρήνον
Κάνει, ενώ σε χαιρετά,
Η Πατρίδα των Ελλήνων
Κλαίγε, κλαίγε, Ελευθεριά.
138
Γιατί εκείτεταν 'ς την κλίνη,
Και του εβάραινε πολύ
Πως για πάντα είχε να μείνη,
Και από Σε να χωριστή·
139 [61]
Αρχινάει του ξεσκεπάζει
Άλλον κόσμο ο λογισμός,
Και κάθε άλλο σκοταδιάζει,
Και του κρύβεται απ' εμπρός.
140
Αλλά αντίκρυ από τα πλάσματα
Του νοός τα αληθινά,
Του προβαίνουν δυο φαντάσματα
ολοζώντανα και ορθά·
141
Η ακριβή του θυγατέρα,
Καθώς έμεινε μικρή, (20)](https://image.slidesharecdn.com/2625997-250609211226-62fa60f1/85/Elliptic-Integrable-Systems-A-Comprehensive-Geometric-Interpolation-Idrisse-Khemar-79-320.jpg)
Elliptic Integrable Systems A Comprehensive Geometric Interpolation Idrisse Khemar
- 1. Elliptic Integrable Systems A Comprehensive Geometric Interpolation Idrisse Khemar download https://ebookbell.com/product/elliptic-integrable-systems-a- comprehensive-geometric-interpolation-idrisse-khemar-5251994 Explore and download more ebooks at ebookbell.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Discrete Integrable Systems Qrt Maps And Elliptic Surfaces 1st Edition Jj Duistermaat Auth https://ebookbell.com/product/discrete-integrable-systems-qrt-maps- and-elliptic-surfaces-1st-edition-jj-duistermaat-auth-2264554 Multiple Integrals In The Calculus Of Variations And Nonlinear Elliptic Systems Am105 Volume 105 Mariano Giaquinta https://ebookbell.com/product/multiple-integrals-in-the-calculus-of- variations-and-nonlinear-elliptic-systems-am105-volume-105-mariano- giaquinta-51954756 Elliptic Integrals And Elliptic Functions Takashi Takebe https://ebookbell.com/product/elliptic-integrals-and-elliptic- functions-takashi-takebe-50834846 Elliptic Integrals Elliptic Functions And Modular Forms In Quantum Field Theory 1st Ed Johannes Blmlein https://ebookbell.com/product/elliptic-integrals-elliptic-functions- and-modular-forms-in-quantum-field-theory-1st-ed-johannes- blmlein-9964438
- 3. Lectures On Selected Topics In Mathematical Physics Elliptic Functions And Elliptic Integrals Schwalm https://ebookbell.com/product/lectures-on-selected-topics-in- mathematical-physics-elliptic-functions-and-elliptic-integrals- schwalm-26906716 On The Direct Numerical Calculation Of Elliptic Functions And Integrals Reprint Edition Louis V King Louis V https://ebookbell.com/product/on-the-direct-numerical-calculation-of- elliptic-functions-and-integrals-reprint-edition-louis-v-king- louis-v-5429464 Elliptic Extensions In Statistical And Stochastic Systems Makoto Katori https://ebookbell.com/product/elliptic-extensions-in-statistical-and- stochastic-systems-makoto-katori-48493824 Elliptic Partial Differential Equations Existence And Regularity Of Distributional Solutions Lucio Boccardo Gisella Croce https://ebookbell.com/product/elliptic-partial-differential-equations- existence-and-regularity-of-distributional-solutions-lucio-boccardo- gisella-croce-50378576 Elliptic Curves A Computational Approach Susanne Schmitt Horst G Zimmer Attila Peth https://ebookbell.com/product/elliptic-curves-a-computational- approach-susanne-schmitt-horst-g-zimmer-attila-peth-50378698
- 5. MEMOIRS of the American Mathematical Society Number 1031 Elliptic Integrable Systems: A Comprehensive Geometric Interpretation Idrisse Khemar September 2012 • Volume 219 • Number 1031 (fourth of 5 numbers) • ISSN 0065-9266 American Mathematical Society
- 6. Number 1031 Elliptic Integrable Systems: A Comprehensive Geometric Interpretation Idrisse Khemar September 2012 • Volume 219 • Number 1031 (fourth of 5 numbers) • ISSN 0065-9266
- 7. Library of Congress Cataloging-in-Publication Data Khemar, Idrisse, 1979- Elliptic integrable systems: a comprehensive geometric interpretation / Idrisse Khemar. p. cm. — (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 1031) “September 2012, volume 219, number 1031 (fourth of 5 numbers).” Includes bibliographical references and index. ISBN 978-0-8218-6925-3 (alk. paper) 1. Geometry, Riemannian. 2. Hermitian structures. I. Title. QA649.K44 2011 516.373—dc23 2012015562 Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Publisher Item Identifier. The Publisher Item Identifier (PII) appears as a footnote on the Abstract page of each article. This alphanumeric string of characters uniquely identifies each article and can be used for future cataloguing, searching, and electronic retrieval. Subscription information. Beginning with the January 2010 issue, Memoirs is accessi- ble from www.ams.org/journals. The 2012 subscription begins with volume 215 and consists of six mailings, each containing one or more numbers. Subscription prices are as follows: for pa- per delivery, US$772 list, US$617.60 institutional member; for electronic delivery, US$679 list, US$543.20 institutional member. Upon request, subscribers to paper delivery of this journal are also entitled to receive electronic delivery. If ordering the paper version, subscribers outside the United States and India must pay a postage surcharge of US$69; subscribers in India must pay a postage surcharge of US$95. Expedited delivery to destinations in North America US$61; else- where US$167. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number. Back number information. For back issues see www.ams.org/bookstore. Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box 845904, Boston, MA 02284-5904 USA. All orders must be accompanied by payment. Other correspondence should be addressed to 201 Charles Street, Providence, RI 02904-2294 USA. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to reprint-permission@ams.org. Memoirs of the American Mathematical Society (ISSN 0065-9266) is published bimonthly (each volume consisting usually of more than one number) by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2294 USA. Periodicals postage paid at Providence, RI. Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA. c 2012 by the American Mathematical Society. All rights reserved. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. This publication is indexed in Mathematical Reviews R , Zentralblatt MATH, Science Citation Index R , SciSearch R , Research Alert R , CompuMath Citation Index R , Current Contents R /Physical, Chemical Earth Sciences. This publication is archived in Portico and CLOCKSS. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 17 16 15 14 13 12
- 8. Contents Introduction 1 0.1. The primitive systems 1 0.2. The determined case 2 0.2.1. The minimal determined system 2 0.2.2. The general structure of the maximal determined case 3 0.2.3. The model system in the even case 3 0.2.4. The model system in the odd case 3 0.2.5. The coupled model system 7 0.2.6. The general maximal determined odd system (k = 2k + 1, m = 2k) 7 0.2.7. The general maximal determined even system (k = 2k, m = 2k − 1) 9 0.2.8. The intermediate determined systems 11 0.3. The underdetermined case 11 0.4. In the twistor space 12 0.5. Related subjects and works, and motivations 13 0.5.1. Relations with surface theory 13 0.5.2. Relations with mathematical physics 13 0.5.3. Relations of F-stringy harmonicity and supersymmetry 14 Notation, conventions and general definitions 15 0.6. List of notational conventions and organisation of the paper 15 0.7. Almost complex geometry 16 Chapter 1. Invariant connections on reductive homogeneous spaces 19 1.1. Linear isotropy representation 19 1.2. Reductive homogeneous space 19 1.3. The (canonical) invariant connection 20 1.4. Associated covariant derivative 20 1.5. G-invariant linear connections in terms of equivariant bilinear maps 21 1.6. A family of connections on the reductive space M 23 1.7. Differentiation in End(T(G/H)) 24 Chapter 2. m-th elliptic integrable system associated to a k -symmetric space 27 2.0.1. Definition of Gτ (even when τ does not integrate in G) 27 2.1. Finite order Lie algebra automorphisms 28 2.1.1. The even case: k = 2k 28 2.1.2. The odd case: k = 2k + 1 30 2.2. Definitions and general properties of the m-th elliptic system 31 iii
- 9. iv CONTENTS 2.2.1. Definitions 31 2.2.2. The geometric solution 33 2.2.3. The increasing sequence of spaces of solutions: (S(m))m∈N 36 2.2.4. The decreasing sequence (Syst(m, τp ))p/k 38 2.3. The minimal determined case 38 2.3.1. The even minimal determined case: k = 2k and m = k 39 2.3.2. The minimal determined odd case 42 2.4. The maximal determined case 45 Adding holomorphicity conditions; the intermediate determined systems 46 2.5. The underdetermined case 47 2.6. Examples 47 2.6.1. The trivial case: the 0-th elliptic system associated to a Lie group 47 2.6.2. Even determined case 47 2.6.3. Primitive case 48 2.6.4. Underdetermined case 48 2.7. Bibliographical remarks and summary of the results 48 Chapter 3. Finite order isometries and twistor spaces 51 3.1. Isometries of order 2k with no eigenvalues = ±1 52 3.1.1. The set of connected components in the general case 52 3.1.2. Study of AdJ, for J ∈ Za 2k(R2n ) 54 3.1.3. Study of AdJj 56 3.2. Isometries of order 2k + 1 with no eigenvalue = 1 58 3.3. The effect of the power maps on the finite order isometries 58 3.4. The twistor spaces of a Riemannian manifolds and its reductions 60 3.5. Return to an order 2k automorphism τ : g → g 60 3.5.1. Case r = k 60 3.5.2. Action of Adτ|m on adgC j 61 3.6. The canonical section in (Z2k(G/H))2 , the canonical embedding, and the twistor lifts 62 3.6.1. The canonical embedding 62 3.6.2. The twistor lifts 63 3.7. Bibliographical remarks and summary of the results 64 Chapter 4. Vertically harmonic maps and harmonic sections of submersions 65 4.1. Definitions, general properties and examples 65 4.1.1. The vertical energy functional 65 4.1.2. Examples 65 4.1.3. Ψ-torsion, Ψ-difference tensor, and curvature of a Pfaffian system 70 4.2. Harmonic sections of homogeneous fibre bundles 72 4.2.1. Definitions and geometric properties 73 4.2.2. Vertical harmonicity equation 76 4.2.3. Reductions of homogeneous fibre bundles 78 4.3. Examples of homogeneous fibre bundles 81 4.3.1. Homogeneous spaces fibration 81 4.3.2. The twistor bundle of almost complex structures Σ(E) 86 4.3.3. The twistor bundle Z2k(E) of a Riemannian vector bundle 89 4.3.4. The twistor subbundle Zα 2k,j(E) 92 4.4. Geometric interpretation of the even minimal determined system 99
- 10. CONTENTS v 4.4.1. The injective morphism of homogeneous fibre bundle 99 4.4.2. Conclusion 102 4.5. Bibliographical remarks and summary of the results 103 Chapter 5. Generalized harmonic maps 105 5.1. Affine harmonic maps and holomorphically harmonic maps 105 5.1.1. Affine harmonic maps: general properties 105 5.1.2. Holomorphically harmonic maps 106 5.2. The sigma model with a Wess-Zumino term in nearly Kähler manifolds 112 5.2.1. Totally skew-symmetric torsion 112 5.2.2. The general case of an almost Hermitian manifold 114 5.2.3. The example of a 3-symmetric space 116 5.2.4. The good geometric context/setting 117 5.2.5. J-twisted harmonic maps 119 5.3. The sigma model with a Wess-Zumino term in G1-manifolds 119 5.3.1. TN-valued 2-forms 119 5.3.2. Stringy harmonic maps 121 5.3.3. Almost Hermitian G1-manifolds 122 5.3.4. Characterization of Hermitian connections in terms of their torsion 126 5.3.5. The example of a naturally reductive homogeneous space 127 5.3.6. Geometric interpretation of the maximal determined odd case 128 5.4. Stringy harmonicity versus holomorphic harmonicity 129 5.5. Bibliographical remarks and summary of the results 130 Chapter 6. Generalized harmonic maps into f-manifolds 133 6.1. f-structures: General definitions and properties 133 6.1.1. f-structures, Nijenhuis tensor and natural action on the space of torsions T 133 6.1.2. Introducing a linear connection 134 6.2. The f-connections and their torsion 135 6.2.1. Definition, notation and first properties 135 6.2.2. Characterization of metric connections preserving the splitting 136 6.2.3. Characterization of metric f-connections. Existence of a characteristic connection 140 6.2.4. Precharacteristic and paracharacteristic connections 146 6.2.5. Reductions of f-manifolds 148 6.3. f-connections on fibre bundles 151 6.3.1. Riemannian submersion and metric f-manifolds of global type G1 151 6.3.2. Reductions of f-submersions 155 6.3.3. Horizontally Kähler f-manifolds and horizontally projectible f-submersions 157 6.3.4. The example of a naturally reductive homogeneous space 158 6.3.5. The example of the twistor space Zα 2k(M) 159 6.3.6. The example of the twistor space Zα 2k,j(M, Jj) 160 6.3.7. The reduction of homogeneous fibre bundle IJ0 : G/G0 → Zα0 2k,2(G/H, J2) 161 6.4. Stringy harmonic maps in f-manifolds 161 6.4.1. Definitions 161
- 11. vi CONTENTS 6.4.2. The closeness of the 3-forms F • T and F T 163 6.4.3. The sigma model with a Wess-Zumino term in reductive metric f-manifold of global type G1 167 6.4.4. The example of a naturally reductive homogeneous space 168 6.4.5. Geometric interpretation of the maximal determined even case 168 6.4.6. Twistorial geometric interpretation of the maximal determined even case 170 6.4.7. About the variational interpretation in the twistor spaces 171 6.5. Bibliographical remarks and summary of the results 172 Chapter 7. Generalized harmonic maps into reductive homogeneous spaces 175 7.1. Affine harmonic maps into reductive homogeneous spaces 176 Affine harmonic maps into symmetric spaces 178 7.2. Affine/holomorphically harmonic maps into 3-symmetric spaces 178 7.3. (Affine) vertically (holomorphically) harmonic maps 179 7.3.1. Affine vertically harmonic maps: general properties 179 7.3.2. Affine vertically holomorphically harmonic maps 180 7.4. Affine vertically harmonic maps into reductive homogeneous space 180 The Riemannian case 182 7.5. Harmonicity vs. vertical harmonicity 184 7.6. (Affine) vertically (holomorphically) harmonic maps into reductive homogeneous space with an invariant Pfaffian structure 186 7.7. The intermediate determined systems 192 7.7.1. The odd case 192 7.7.2. The even case 193 7.7.3. Sigma model with a Wess-Zumino term 194 7.8. Some remarks about the twistorial interpretation 195 7.8.1. The even case 195 7.8.2. The odd case 195 7.9. Bibliographical remarks and summary of the results 196 Chapter 8. Appendix 197 8.1. Vertical harmonicity 197 8.2. G-invariant metrics 199 8.2.1. About the natural reductivity 199 8.2.2. Existence of an AdH-invariant inner product on k for which τ|m is an isometry 200 8.2.3. Existence of a naturally reductive metric for which J is an isometry, resp. F is metric 201 Bibliography 203 Index 207 List of symbols 209 Section 1 209 Section 2 210 Section 3 212 Section 4 214 Section 5 215
- 12. CONTENTS vii Section 6 216 Section 7 217
- 14. Abstract In this paper, we study all the elliptic integrable systems, in the sense of C. L. Terng [66], that is to say, the family of all the m-th elliptic integrable systems associated to a k -symmetric space N = G/G0. Here m ∈ N and k ∈ N∗ are integers. For example, it is known that the first elliptic integrable system asso- ciated to a symmetric space (resp. to a Lie group) is the equation for harmonic maps into this symmetric space (resp. this Lie group). Indeed it is well known that this harmonic maps equation can be written as a zero curvature equation: dαλ + 1 2 [αλ ∧ αλ] = 0, ∀λ ∈ C∗ , where αλ = λ−1 α 1 + α0 + λα 1 is a 1-form on a Riemann surface L taking values in the Lie algebra g. This 1-form αλ is obtained as follows. Let f : L → N = G/G0 be a map from the Riemann sur- face L into the symmetric space G/G0. Then let F : L → G be a lift of f, and consider α = F−1 .dF its Maurer-Cartan form. Then decompose α according to the symmetric decomposition g = g0 ⊕ g1 of g : α = α0 + α1. Finally, we define αλ := λ−1 α 1+α0+λα 1 , ∀λ ∈ C∗ , where α 1, α 1 are the resp. (1, 0) and (0, 1) parts of α1. Then the zero curvature equation for this αλ, for all λ ∈ C∗ , is equivalent to the harmonic maps equation for f : L → N = G/G0, and is by definition the first ellip- tic integrable system associated to the symmetric space G/G0. Thus the methods of integrable system theory apply to give generalised Weierstrass representations, algebro-geometric solutions, spectral deformations, and so on. In particular, we can apply the DPW method [23] to obtain a generalised Weierstrass representa- tion. More precisely, we have a Maurer-Cartan equation in some loop Lie algebra Λgτ = {ξ : S1 → g|ξ(−λ) = τ(ξ(λ))}. Then we can integrate it in the corresponding loop group and finally apply some factorization theorems in loop groups to obtain a generalised Weierstrass representation: this is the DPW method. Moreover, these methods of integrable system theory hold for all the systems written in the forms of a zero curvature equation for some αλ = λ−m α̂−m +· · ·+α̂0 +· · ·+λm α̂m. Namely, these methods apply to construct the solutions of all the m-th elliptic integrable systems. So it is natural to ask what is the geometric interpretation of these sys- tems. Do they correspond to some generalisations of harmonic maps? This is the problem that we solve in this paper: to describe the geometry behind this family Received by the editor April 8, 2010 and, in revised form, March 31, 2011. Article electronically published on February 22, 2012; S 0065-9266(2012)00651-4. 2010 Mathematics Subject Classification. Primary 53B20, 53B35; Secondary 53B50. Research supported successively by the DFG-Schwerpunkt Globale Differentialgeometrie, grant DO 776 (in T.U. Munich), and then by the SFB - TR 71 (in Universität Tübingen). Affiliation at time of publication: Institut Élie Cartan de Nancy, Université Henri Poincaré Nancy 1, B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cdex, France; email: khemar@iecn.u-nancy.fr. c 2012 American Mathematical Society ix
- 15. x ABSTRACT of integrable systems for which we know how to construct (at least locally) all the solutions. The introduction below gives an overview of all the main results, as well as some related subjects and works, and some additional motivations.
- 16. Introduction In this paper, we give a geometric interpretation of all the m-th elliptic in- tegrable systems associated to a k -symmetric space N = G/G0 (in the sense of C.L. Terng [66]). Let g be a real Lie algebra and τ : g → g be an automorphism of finite order k . This automorphism admits an eigenspace decomposition gC = ⊕j∈Zk gC j , where gC j is the eigenspace of τ with respect to the eigenvalue ωj k . We denote by ωk a k -th primitive root of unity. Moreover the automorphism τ defines a k -symmetric space N = G/G0. Furthermore let L be a Riemann surface. The m-th elliptic integrable system associated to (g, τ) can be written as a zero curvature equation dαλ + 1 2 [αλ ∧ αλ] = 0, ∀λ ∈ C∗ , where αλ = m j=0 λ−j uj +λj ūj = m j=−m λj α̂j is a 1-form on the Riemann surface L taking values in the Lie algebra g. The ”coefficient” uj is a (1, 0)-type 1-form on L with values in the eigenspace gC −j. Moreover, we call the integer m the order of the system. Let us make precise that the order m has nothing to do with the order of any PDE, but this is only the maximal power on λ in the (finite) Fourier decomposition of αλ w.r.t. λ. First, we remark that any solution of the system of order m is a solution of the system of order m , if m ≤ m (and the automorphism τ is fixed). In other words, the system of order m is a reduction of the system of order m , if m ≤ m . Moreover, it turns out that we have to introduce the integer mk defined by mk = k + 1 2 = k if k = 2k k + 1 if k = 2k + 1 if k 1, and m1 = 0. Then the general problem splits into three cases : the primitive case (m mk ), the determined case (mk ≤ m ≤ k − 1) and the underdetermined case (m ≥ k ). 0.1. The primitive systems. The primitive systems have an interpretation in terms of F-holomorphic maps, with respect to an f-struture F on the target space N = G/G0 (i.e. an endomor- phism F satisfying F3 + F = 0). More precisely: • In the even case (k = 2k), we have a fibration π: G/G0 → G/H over a k-symmetric space M = G/H (defined by the square τ2 of the automorphism τ of order k defining N = G/G0). We also have a G-invariant splitting TN = H ⊕ V corresponding to this fibration1 , where V = ker dπ. Moreover N is naturally endowed with an f-structure F which defines a complex structure on the hori- zontal subbundle H and vanishes on the vertical subbundle V. Furthermore, the 1i.e. H is a connection on this fibration. 1
- 17. 2 IDRISSE KHEMAR eigenspace decomposition of the order k automorphism τ gives us some G-invariant decomposition H = ⊕k−1 j=1 [mj], where mj ⊂ g is defined by mC j = gC −j ⊕ gC j , and [mj] ⊂ TN is the corresponding G-invariant subbundle. This allows to define, by multiplying F on the left by the projections on the subbundles Hm = ⊕m j=1[mj], a family of f-structures F[m] , 1 ≤ m ≤ k − 1. Then the primitive system of order m (m mk = k) associated to G/G0 is exactly the equation for F[m] - holomorphic maps. Therefore the solutions of the primitive systems are exactly the F-holomorphic maps. • In the odd case (k = 2k+1), N = G/G0 is naturally endowed with an almost complex structure J. Then the solutions of the primitive systems are exactly the J- holomorphic curves. Moreover, in the same way as for the even case, the eigenspace decomposition of τ provides a G-invariant decomposition TN = ⊕k j=1[mj], which allows to define a family of f-structures F[m] , 1 ≤ m ≤ k, with F[k] = J. Then the primitive system of order m (m mk = k + 1) associated to G/G0 is exactly the equation for F[m] -holomorphic maps. In other words, the solutions of the primitive system of order m are exactly the integral holomorphic curves of the complex Pfaffian system ⊕m j=1[mj] ⊂ TN in the almost complex manifold (N, J). 0.2. The determined case. We call “the minimal determined system” the determined system of minimal or- der mk , and “the maximal determined system” the determined system of maximal order k − 1. Any solution of a determined system is solution of the corresponding maximal determined system. More precisely, a map f : L → G/G0 is solution of a deter- mined system (associated to G/G0) if and only if it is solution of the maximal determined system (associated to G/G0) and satisfies an additional holomorphicity condition. When this holomorphicity condition is maximal, then we obtain the minimal determined system. 0.2.1. The minimal determined system. The minimal determined system has an interpretation in terms of horizontally holomorphic and vertically harmonic maps f : L → N = G/G0. It also has an equivalent interpretation in terms of vertically harmonic twistor lifts in some twistor space. Let us make precise this point. In the even case. As we have seen in the subsection 0.1 below, the homoge- neous space N = G/G0 admits a G-invariant splitting TN = H ⊕ V corresponding to the fibration π: N → M and N is naturally endowed with an f-structure F which defines a complex structure on the horizontal subbundle H and vanishes on the vertical subbundle V. Then we say that a map f : L → N is horizontally holomorphic if (df ◦ jL)H = F ◦ df. Then we prove that the even minimal determined system (Syst(k, τ)) means that the geometric map f is horizontally holomorphic and vertically harmonic, i.e. τv (f) := Trg(∇v dv f) = 0 (for any hermitian metric g on the Riemann surface L). Here ∇v is the vertical component of the Levi-Civita connection ∇ (of some G-invariant metric on N). Moreover, vertically harmonic maps are exactly the critical points (w.r.t. vertical
- 18. INTRODUCTION 3 variations) of the vertical energy functional: Ev (u) = 1 2 L |dv u|2 dvolg. We prove also that this system also has an equivalent interpretation in terms of vertically harmonic twistor lifts in some twistor space Z2k,j(M, J2) which is a subbundle of Z2k(M), where Zk (M) = {J ∈ SO(TM)| Jk = Id, Jp = Id if p k , ker(J ± Id) = {0}} is the bundle of isometric endomorphisms of TM with finite order k and with no eigenvalues = ±1. More precisely denoting by J2 the section of SO(TM), of order k, defined by τ2 m, then we define Z2k,j(M, J2) = {J ∈ Z2k(M)|J2 = J2}. Then we prove that N = G/G0 can be embedded into the twistor space Z2k,j(M, J2) via a natural morphism of bundle over M = G/H. We prove that f : L → N is solution of the system if and only if the corresponding map Jf : L → Z2k,j(M, J2) is a vertically harmonic twistor lift. In the odd case. We obtain an analogous interpretation as in the even case. An interpretation in terms of horizontally holomorphic and vertically harmonic maps f : L → N = G/G0. Let us make precise that in the odd case, the action functional in the variational interpretation has a Wess-Zumino term in addition to the vertical energy (See below in this introduction). Moreover by embedding G/G0 into the twistor space Z2k+1(N) of order 2k + 1 isometric endomorphisms in TN, we obtain an interpretation in terms of vertically harmonic twistor lift. 0.2.2. The general structure of the maximal determined case. First, the maxi- mal determined system has 3 model cases. This means that we can distinguish 3 maximal determined systems, namely the three maximal determined systems with lowest order of symmetry (2,3,4). Their corresponding geometric equations (when put all together) contain already all the structure terms - in a simple form- that will appear in the further maximal determined systems in a more complex and general form due to the more complex geometric structure in the further maximal determined systems. That is in this sense that we can say that all the further determined systems associated to target spaces N with higher order of symmetry will be modeled on these model systems. 0.2.3. The model system in the even case. In the even case, this model is the first elliptic integrable system associated to a symmetric space (m = 1, k = 2) which is - as it is well known - exactly the equation of harmonic maps from the Riemann surface L into the symmetric space under consideration. This is the “smallest” determined system, i.e. with lowest order of symmetry in the target space N = G/G0. In this case -N is symmetric- the determined case is reduced to one system, the one of order 1. 0.2.4. The model system in the odd case. In the odd case, this model is the second elliptic integrable system associated to a 3-symmetric space. This is the “smallest” determined system in the odd case, i.e. with lowest odd order of sym- metry in the target space N = G/G0. We prove that this system is exactly the equation for holomorphically harmonic maps into the almost complex manifold (N, J) with respect to the anticanonical connection ∇1 = ∇0 + [ , ][m], where ∇0
- 19. 4 IDRISSE KHEMAR is the canonical connection. Or equivalently this is the equation for holomorphi- cally harmonic maps into the almost complex manifold (N, −J) with respect to the canonical connection ∇0 . Holomorphically harmonic maps. Given a general almost complex mani- fold (N, J) with a connection ∇, we define holomorphically harmonic maps f : L → N as the solutions of the equation (0.1) ¯ ∂∇ ∂f 1,0 = 0 where [ ]1,0 denotes the (1, 0)-component according to the splitting TNC = T1,0 N ⊕ T0,1 N defined by J. This equation is equivalent to d∇ df + Jd∇ ∗ df = 0 or, equivalently, using any Hermitian metric g on L Tg(f) + Jτg(f) = 0 where Tg(f) = ∗f∗ T = f∗ T(e1, e2), with (e1, e2) an orthonormal basis of TL, and τg(f) = ∗d∇ ∗ df = Trg(∇df) is the tension field of f. Of course Trg denotes the trace with respect to g, and the expression ∇df denotes the covariant derivative of df with respect to the connection induced in T∗ L⊗f∗ TN by ∇ and the Levi-Civita connection in L. In particular, we see that if ∇ is torsion free or more generally if f is torsion free, i.e. f∗ T = 0, then holomorphic harmonicity is equivalent to (affine) harmonic- ity. Therefore, this new notion is interesting only in the case of a nontorsion free connection ∇. The vanishing of some ¯ ∂∂-derivative. Now, let us suppose that the con- nection ∇ on N is almost complex, i.e. ∇J = 0. Then, according to equation (0.1), we see that any holomorphic curve f : (L, jL) → (N, J) is anti-holomorphically harmonic, i.e. holomorphically harmonic with respect to −J. In particular, this allows to recover that a 1-primitive solution (i.e. of order m = 1) of the elliptic sys- tem associated to a 3-symmetric space is also solution of the second elliptic system associated to this space. Moreover, the holomorphically harmonic maps admit a formulation very anal- ogous to that of harmonic maps in terms of the vanishing of some ¯ ∂∂-derivative, which implies a well kown characterisation in terms of holomorphic 1-forms. Indeed we prove that f : (L, jL) → (N, J, ∇) is holomorphically harmonic if and only if (0.2) ˆ ∂ ∇ ˆ ∂f = 0, i.e. ˆ ∂f is a holomorphic section of T∗ 1,0L⊗C f∗ TN. Here the hat “ˆ” means that we extend a 1-form on TL, like d or ∇, by C-linearity as a linear map from TLC into the complex bundle (TN, J). In other words instead of extending these 1-forms as C-linear maps from TLC into TNC as it is usual, we use the already existing structure of complex vector bundle in (TN, J) and extend these very naturally as C- linear map from TLC into the complex bundle (TN, J). Therefore we can conclude that holomorphically harmonic maps have the same formulation as harmonic maps with the difference that instead of working in the complex vector bundle TNC , we stay in TN which is already a complex vector bundle in which we work.
- 20. INTRODUCTION 5 The sigma model with a Wess-Zumino term. Finally, let us suppose that N is endowed with a ∇-parallel Hermitian metric h. Therefore (N, J, h) is an almost Hermitian manifold with a Hermitian connection ∇. Suppose also that J anticommutes with the torsion T of ∇ i.e. T(X, JY ) = −JT(X, Y ) which is equivalent to T = 1 4 NJ where NJ denotes the torsion of J i.e its Nijenhuis tensor. Suppose also that the torsion of ∇ is totally skew-symmetric i.e. the trilinear map T∗ (X, Y, Z) = T(X, Y ), Z is a 3-form. Lastly, we suppose that the torsion is ∇-parallel, i.e. ∇T∗ = 0 which is equivalent to ∇T = 0. Then we prove that this implies that the 3-form H(X, Y, Z) = −T∗ (X, Y, JZ) = JT(X, Y ), Z is closed dH = 0.2 Then the equation for holomorphically harmonic maps f : L → N is the equa- tion of motion (i.e. Euler-Lagrange equation) for the sigma model in N with the Wess-Zumino term defined by the closed 3-form H. The action functional is given by S(f) = E(f) + SW Z (f) = 1 2 L |df|2 dvolg + B H, where B is 3-submanifold (or indeed a 3-chain) in N whose boundary is f(L). Then since dH = 0, the variation of the Wess-Zumino term is a boundary term δSW Z = B Lδf H = B dıδf H = f(L) ıδf H, whence its contribution to the Euler-Lagrange equation involves only the original map f : L → N. In particular, applying this result to the case we are interested in, i.e. N is 3-symmetric, we obtain: The second elliptic system associated to a 3-symmetric space N = G/G0 is the equation of motion for the sigma model in N with the Wess-Zumino term defined by the closed 3-form H(X, Y, Z) := T∗ (X, Y, JZ), where T is the torsion of the canonical connection ∇0 and J is the canonical almost complex structure.3 2Let us point out that in general T∗ is not closed even if it is ∇-parallel. For example, in a Riemannian naturally reductive homogeneous space G/H, endowed with its canonical con- nection ∇0, we have ∇0T = 0 but dT∗(X, Y, Z, V ) = −2Jacm(X, Y, Z), V where Jacm is the m-component of the Jacobi identity (i.e. the sum of the circular permutations of [X, [Y, Z]m]m). Of course m denotes the AdH-invariant summand in the reductive decomposition g = h ⊕ m. 3In fact, we need a naturally reductive metric on N to ensure that T∗ is a 3-form. But if we allow Pseudo-Riemannian metrics and if g is semisimple then the metric defined by the Killing form is naturally reductive. In fact, the elliptic integrable system is a priori written in an affine context, i.e. its natural - in the sense of initial- geometric interpretation takes place in the context of affine geometry in terms of the linear connections ∇t = ∇0 + t[ , ][m]. If we want that this interpretation takes place in the context of Riemannian geometry we need, of course, to add some hypothesis of compactness, like the compactness of AdmG0 and the natural reductivity. But we do not need these hypothesis if we work in the Pseudo-Riemannian context.
- 21. 6 IDRISSE KHEMAR The good geometric context/setting. In the previous variational interpre- tation, we need to make 3 hypothesis on the torsion of the Hermitian connection: T anticommutes with J, is totally skew-symmetric and ∇-parallel. It is natural to ask ourself what do these hypothesis mean geometrically and what is the good geometric context in which these take place. It turns out that the good geometric context is the one of nearly Kähler manifold. An almost Hermitian manifold (N, J, h) is nearly Kähler if and only if (∇h XJ)X = 0, for all X ∈ TN, where ∇h is the Levi-Civita connection. Then we prove that the almost Hermitian manifolds for which there exists an Hermitian connection satisfiying the three hypothesis above are exactly Nearly Kähler manifolds, and that this Hermitian connection is then unique and coincides with the canonical Hermitian connection. Then the variational interpretation can be rewritten as follows: Theorem 0.2.1. Let (N, h, J) be a nearly Kähler manifold then the equation of holomorphic harmonicity, w.r.t. the canonical Hermitian connection, for maps f : L → N is exactly the Euler-Lagrange equation for the sigma model in N with a Wess-Zumino term defined by the 3-form: H = 1 3 dΩJ where ΩJ = J·, · is the Kähler form. Therefore: the second elliptic system associated to a 3-symmetric space N = G/G0, endowed with its canonical almost complex structure J, is the equation of motion for the sigma model in N with the Wess-Zumino term defined by the closed 3-form H = − 1 3 dΩJ. J-twisted harmonic maps. We prove that we can also interpret the holo- morphic harmonicity in terms of J-twisted harmonic maps (w.r.t. the Levi-Civita connection). Let us define this notion. Let (E, J) be a complex vector bundle over an almost complex manifold (M, jM ). Then let ∇ be a connection on E. Then we can decompose it in an unique way as the sum of a J-commuting and a J-anticommuting part, i.e. in the form ∇ = ∇0 + A where ∇0 J = 0 and A ∈ C(T∗ M ⊗ End(E)), AJ = −JA. More precisely, we have A = 1 2 J∇J. Then we set ∇ J = ∇0 − (A ◦ jM )J = ∇ − 1 2 J∇J − 1 2 ∇J ◦ jM . Now let f : (L, jL) → (N, J) be a map from a Riemann surface into the almost complex manifold (N, J) endowed with a connection ∇. Then let us take in what precede (M, jM ) = (L, jL) and (E, ∇) = (f∗ TN, f∗ ∇). Then we say that the map f : (L, jL) → (N, J, ∇) is J-twisted harmonic if and only if Trg(∇ J df) = 0 (for any hermitian metric g on the Riemann surface L).
- 22. INTRODUCTION 7 0.2.5. The coupled model system. This is the third elliptic integrable system associated to a 4-symmetric space. The corresponding geometric equation in the 4-symmetric space G/G0 can be viewed as a coupling between the equation of harmonic maps into the symmetric fibre H/G0 and the equation for harmonic maps into the symmetric space G/H. In other words, it can be viewed as a coupling between the even model system associated to the symmetric space G/H and the even model system associated to the symmetric space H/G0. In particular, if harmonicity is heuristically replaced by holomorphicity ”in the basis G/H”, then we recover the horizontally holomorphic and vertically harmonic maps into G/G0, that is to say the solutions of the second elliptic integrable system associated to G/G0. We will come back to this in the end of section 0.2.7 in this introduction. 0.2.6. The general maximal determined odd system (k = 2k +1, m = 2k). The maximal determined odd system has a geometric interpretation in terms of stringy harmonic maps f : L → (G/G0, J), with respect to the canonical connection and the canonical almost complex structure. Stringy harmonic maps. Let (N, J) be an almost complex manifold with ∇ an linear connection then we will say that a map f : L → N from a Riemann surface into N is stringy harmonic if it is a solution of the harmonic map equation with a JT-term: −τg(f) + (J · T)g(f) = 0. We have used the notation J·B = −JB(J·, ·), ∀B ∈ C(Λ2 T∗ N⊗TN). This action of J on B ∈ C(Λ2 T∗ N ⊗TN) can be written more naturally if (N, J) is endowed with a Hermitian metric h. Indeed, in this case, we have an identification, C(Λ2 T∗ N ⊗ TN) = C(Λ2 T∗ N ⊗ T∗ N) ⊂ C(⊗3 T∗ N), between TN-valued 2-forms on N and trilinear forms on N skew-symmetric w.r.t. the 2 first variables: B(X, Y, Z) := B(X, Y ), Z. Then J · B is written: J · B = B(J·, J·, J·) =: −Bc . We remark that if T anticommutes with J then stringy harmoniciy coincides with holomorphic harmonicity (since in this case J ·T = JT). More particulary, if T = 0, then the stringy harmonicity coincides with the harmonicity. Furthermore, we look for a general geometric setting in which the stringy har- monicity has an interesting interpretation. First of all, let us remark that in the context of homogeneous reductive space, in which our system takes place, we have a canonical connection, with respect to which the stringy harmonicity can be written “canonically”. But in general we do not have a “special” connection with respect to which one can consider the stringy harmonicity. Therefore, if one wants to place stringy harmonicity in a more meaningfull, interesting and fruitful context (than the general context of almost complex manifolds endowed with some linear con- nection) and, in so doing, obtain a better understanding of our elliptic integrable system by writting its geometric interpretation in the best geometric context, a first problem - that we solved - is to find a general class of (almost complex) manifold in which there exists some unique “canonical” connection, with respect to which we then could consider the stringy harmonicity. This provides us, firstly, some special connection (in the same sense that the Levi-Civita connection is special in Riemannian geometry), which solves the problem of the choice of the connection, but secondly it turns out that it provides also a variationnal interpretation of the stringy harmonicity.
- 23. 8 IDRISSE KHEMAR Best geometric setting. It turns out that the more rich geometric context in which stringy harmonicity admits interesting properties is the one of G1-manifolds, more precisely G1-manifolds whose the characteristic connection has a parallel tor- sion. Making systematic use of the covariant derivative of the Kähler form, A. Gray and L. M. Hervella, in the late seventies, classified almost Hermitian struc- tures into sixteen classes [27]. Denote by W the space of all trilinear forms (on some Hermitian vector space, say Ty0 N for some reference point y0 ∈ N) having the same algebraic properties as ∇h ΩJ . Then they proved that we have a U(n)- irreducible decomposition W = W1 ⊕ W2 ⊕ W3 ⊕ W4. The sixteen classes are then respectively the classes of almost Hermitian manifolds for which ∇h ΩJ ‘lies in’ the U(n)-invariant subspaces WI = ⊕i∈I Wi, I ⊂ {1, . . . , 4}, respectively. In particular, if we take as invariant subspace {0}, we obtain the Kähler manifolds, if we take W1, we obtain the class of nearly Kähler manifolds. Moreover the class of G1-manifolds is the one defined by G1 = W1 ⊕ W3 ⊕ W4. It is characterised by : (N, J, h) is of type G1 if and only if the Nijenhuis tensor NJ is totally skew-symmetric (i.e. a 3-form). In this paper, we prove the following theorem4 : Theorem 0.2.2. An almost Hermitian manifold (N, J, h) admits a Hermitian connection with totally skew-symmetric torsion if and only if the Nijenhuis tensor NJ is itself totally skew-symmetric. In this case, the connection is unique and determined by its torsion which is given by T = −dc ΩJ + NJ . The characteristic connection is then given by ∇ = ∇h − 1 2 T. Then we prove: Proposition 0.2.1. Let us suppose that the almost Hermitian manifold (N,J, h) is a G1-manifold. Let us suppose that its characteristic connection ∇ has a parallel torsion ∇T = 0. Then the 3-form H(X, Y, Z) = T(JX, JY, JZ) = (J · T)(X, Y ), Z is closed dH = 0. This then gives us the following variational interpretation. Theorem 0.2.3. Let us suppose that the almost Hermitian manifold (N, J, h) is a G1-manifold. Let us suppose that its characteristic connection ∇ has a parallel torsion ∇T = 0. Then the equation for stringy harmonic maps f : L → N is exactly the Euler- Lagrange equation for the sigma model in N with a Wess-Zumino term defined by the closed 3-form H = −dΩJ + JNJ . Moreover any (2k + 1)-symmetric space (G/G0, J, h) endowed with its canon- ical almost complex structure and a naturally reductive G-invariant metric h (for which J is orthogonal) is a G1-manifold and moreover its characteristic connection coincides with its canonical connection ∇0 . Finally, the torsion of the canonical 4After the fact, we realized that it has already been proved by Friedrich-Ivanov[24]. However we give a different and completely written proof. See remark 5.3.4
- 24. INTRODUCTION 9 connection is obviously parallel. Therefore we obtain an interpretation of the max- imal determined system associated to a (2k + 1)-symmetric space in terms of a sigma model with a Wess-Zumino term. Remark 0.2.1. Let us add about stringy harmonicity that we also prove, in this paper, that the stringy harmonicity w.r.t. an almost complex connection ∇ is equivalent to the holomorphic harmonicity w.r.t. a new almost complex connection ∇ . 0.2.7. The general maximal determined even system (k = 2k, m = 2k − 1). In the even case, the geometric structure of the target space G/G0 is more complex (and more rich): as we have already seen, there is a fibration π: N = G/G0 → M = G/H, a splitting TN = H ⊕ V with V = ker π, and an f-structure F such that ker F = V and Im F = H (in particular ¯ J := FH is a complex structure on H). Moreover the geometric PDE obtained from our elliptic integrable system uses this geometric structure. In particular, this geometric PDE splits into its horizontal and vertical parts and can be viewed as a coupling between the equation of ¯ J-stringy harmonicity and the equation of vertical harmonicity, the coupling terms calling out the curvature of H. The maximal determined even system has a geometric interpretation in terms of stringy harmonic maps f : L → (G/G0, F), F being the canonical f-structure on G/G0. Stringy harmonic maps w.r.t. an f-structure. Let (N, F) be an f-manifold with ∇ a linear connection. Then we will say that a map f : L → N from a Rie- mann surface into N is stringy harmonic if it is solution of the stringy harmonic maps equation: −τg(f) + (F • T)g(f) = 0, where F • B, for B ∈ C(Λ2 T∗ N ⊗ TN), denotes some natural (linear) action of F on C(Λ2 T∗ N ⊗TN). For more simplicity, let us write it in the case where (N, F) is endowed with a compatible metric h (i.e. V ⊥ H and ¯ J = F|H is orthogonal with respect to h|H×H): F • B = B(F·, F·, F·) + 1 2 F · (B − BH3 ) F · A = A(F·, ·, ·) + A(·, F·, ·) + A(·, ·, F·) for all B, A ∈ C(Λ2 T∗ N ⊗ TN). Now, we want to proceed as in the case of stringy harmonicity with respect to an almost complex structure. That is to say: to find a class of f-manifolds for which there exists some unique characteristic connection which preserves the structure and then to look for a variational interpretation of the stringy harmonicity with respect to this connection. Best Geometric context. We look for metric f-manifolds (N, F, h) for which there exists a metric f-connection ∇ (i.e. ∇F = 0 and ∇h = 0) with skew- symmetric torsion T. In a first step, we consider metric connections which preserve the splitting TN = V ⊕ H (i.e. ∇q = 0, where q is the projection on V) and we characterize the manifolds (N, h, q) for which there exists such a connection with skew-symmetric torsion, and call these reductive metric f-manifolds.
- 25. 10 IDRISSE KHEMAR Then, saying about a metric f-manifolds (N, F, h) that it is of global type G1 if its extended Nijenhuis tensor ÑF is skew-symmetric, we prove the following theorem: Theorem 0.2.4. A metric f-manifold (N, F, h) admits a metric f-connection ∇ with skew-symmetric torsion if and only if it is reductive and of global type G1. Moreover, in this case, for any α ∈ C(Λ3 V∗ ), there exists a unique metric connection ∇ with skew-symmetric torsion such that T|Λ3V = α. This unique connection is given by T = (−dc ΩF + NF |H3 ) + Skew(Φ) + Skew(RV ) + α. where ΩF = F·, ·, Φ and RV are resp. the curvature of H and V resp., and Skew is the sum of all the circular permutations on the three variables. On a metric f-manifold (N, F, h), a metric f-connection ∇ with skew-symmetric torsion will be called a characteristic connection. Moreover, we prove that for any reductive metric f-manifold of global type G1, the closure of H = F • T is equivalent to the closure of the 3-form F · NF − 1 2 F · (Skew(Φ) + Skew(RV)), so that: Theorem 0.2.5. Let (N, F, h) be a reductive metric f-manifold of global type G1. Let us suppose that the 3-form F · NF − 1 2 F · (Skew(Φ) + Skew(RV)) is closed, where Φ and RV are respectively the curvatures of the horizontal and vertical sub- bundles. Let ∇ be one characteristic connection. Then the equation for stringy harmonic maps, f : L → N, (w.r.t. ∇) is exactly the Euler-Lagrange equation for the sigma model in N with a Wess-Zumino term defined by the closed 3-form H = −dΩF + F · NF − 1 2 F · (Skew(Φ) + Skew(RV)) . Contrary to the case of stringy harmonic maps into an almost Hermitian G1- manifolds, in the present case, the hypothesis that the torsion of one characteristic connection is parallel ∇T = 0 does not imply the closure of the 3-form H = F • T. However, we characterize this closure under the hypothesis ∇T = 0 and RV = 0, by some 2 conditions that we will not explain in this introduction (see section 6.4.2): the horizontal complex structure ¯ J is a cyclic permutation of the horizontal curvature, and the 2-forms N ¯ J and Φ have orthogonal supports. Theorem 0.2.6. Let (N, F, h) be a reductive metric f-manifold of global type G1. Let us suppose that one of its characteristic connections, ∇, has a parallel torsion ∇T = 0. Let us supppose that RV = 0 and that the horizontal curvature Φ is pure. The following statement are equivalent: • The horizontal 3-form F · NF is closed. • F · NF and F · (Skew(Φ) + Skew(RV)) are closed. • The horizontal complex structure ¯ J is a cyclic permutation of the horizontal curvature, and the 2-forms N ¯ J and Φ have orthogonal supports. In this case, H = F • T is closed. Moreover any 2k-symmetric space (G/G0, F, h) endowed with its canonical f- structure and a naturally reductive G-invariant metric h (compatible with F) is reductive and of global type G1, and moreover its canonical connection ∇0 is a
- 26. INTRODUCTION 11 characteristic connection. Furthermore, the torsion of the canonical connection is obviously parallel. Finally, we prove that any 2k-symmetric space (G/G0, F, h) satisfies the two hypothesis above. Therefore we obtain an interpretation of the maximal determined even system associated to a 2k-symmetric space in terms of a sigma model with a Wess-Zumino term. A particular case: Horizontally Kähler f-manifolds. Let (N, F, h) be a metric f-manifold. We will say that (N, F, h) is horizontally Kähler if DF|H3 = 0, where D is the Levi-Civita connection of h. Then, we prove that in this case, the two hypothesis above (which characterise the closure of the 3-form H) are satisfied. Moreover, any characteristic connection ∇ satisfies TH3 = 0, which leads to special properties. A example of this situation is given by any 4-symmetric space, endowed with its canonical f-structure and a naturally reductive G-invariant metric (compatible with F). 0.2.8. The intermediate determined systems. For the intermediate determined systems (mk m k − 1), these are obtained from the maximal determined case by adding holomorphicity in the subbundle Hm = ⊕ m j=1[mj] ⊂ H, where m = k − 1 − m. It means that the m-th determined system has a geometric interpretation in terms of stringy harmonic maps which are Hm -holomorphic: (df ◦ jL)Hm = F[m] ◦ df. We have seen that the maximal determined system has an interpretation in terms of a sigma model with a Wess-Zumino term defined by a 3-form H. In fact, more generally, let mk ≤ m ≤ k − 1, and let us consider the splitting TN = Hm ⊕ Vm defined above. Then one can prove that any m-th determined system is the Euler- Lagrange equation w.r.t. vertical variations (i.e. in Vm ) of the following functional Ev̄ (f) = 1 2 L |dv̄ f|2 dvolg + B Hv̄ where dv̄ f = [df]Vm , Hv̄ = H − H = H|S(Vm,Hm), H = H|(Hm)3 , and B is a 3- submanifold of N with boundary ∂B = f(L). We have set S(Vm , Hm ) = S(Vm × Hm × Hm ) and S(E1 × E2 × E3) = S i,j,k Ei ⊗ Ej ⊗ Ek for any E1, E2, E3 vector bundles over N. 0.3. The underdetermined case. We prove that the m-th underdetermined system (m k − 1) is in fact equiv- alent to some m-th determined or primitive system associated to some new auto- morphism τ̃ defined in a product gq+1 , of the initial Lie algebra g. More precisely, we write m = qk + r, 0 ≤ r ≤ k − 1 the Euclidean division of m by k . Then we consider the automorphism in gq+1 defined by τ̃ : (a0, a1, . . . , aq) ∈ gq+1 −→ (a1, . . . , aq, τ(a0)) ∈ gq+1 . Then τ̃ is of order (q + 1)k . We prove that the initial m-th (underdetermined) system associated to (g, τ) is in fact equivalent to the m-th (determined) system associated to (gq+1 , τ̃).
- 27. 12 IDRISSE KHEMAR 0.4. In the twistor space. For each previous geometric interpretation in the target space N = G/G0, there is a corresponding geometric interpretation in the twistor space. • These previous geometric interpretations take place in some manifolds en- dowed with some particular structure. This could simply be, for example, a struc- ture of almost complex manifold in the case of the interpretation of the stringy harmonicity in terms of the vanishing of some ¯ ∂∂-derivative but it could also be the more strong structure of G1-manifolds whose characteristic connection has a parallel torsion. Moreover, our k -symmetric spaces are very particular examples of this kind of manifolds. Therefore it is natural to try to make these interpreta- tions more universal by writting them in a more general setting. More precisely we would like to find some universal prototype of these ”special” manifolds, which can be endowed canonically with the needed geometric structure and such that any of our special manifolds can be embedded in this prototype (see also [48]). Indeed, as concerns k -symmetric spaces, we know that they can be embed- ded canonically into some twistor bundles. In the even case we have an injective morphim of bundle over M = G/H defined by the embedding G/G0 → Z2k(G/H), whereas in the odd case we have a chapter defined by the embedding G/G0 → Z2k+1(G/G0). • In the even case, the fibration π: G/G0 → G/H imposes to view canonically any 2k-symmetric space as a subbundle of Z2k(G/H) so that the twistorial interpre- tation is in some sense dictated by the structure of the 2k-symmetric space. The geometric interpretations in the twistor spaces are universal since these twistor spaces are defined for any Riemannian manifold and are endowed canonically with the different geometric structures that we need. That is to say the geometric structures we need to endow the target space N with, in our previous geometric interpretations. More generally, suppose that we want to study stringy harmonicity in metric f-manifolds (N, F, h). It is then natural to consider the particular case where the vertical subbundle V is the tangent space to the fibre of a Riemannian submersion π: (N, h) → (M, g), i.e. V = ker dπ. For example, we have seen that among the sufficient list of conditions for our variationnal interpretation of stringy harmonicity there is the condition RV = 0. In the particular case of a Riemannian submersion, the f-structure F defines a complex structure ¯ J on π∗ TM = H which itself gives rise to a morphism of submersion I : N → Σ(M), y → (π(y), ¯ J(y)). This shows that the twistor bundle Σ(M) appears naturally in the general context - even though the morphism I is not injective in general. Furthermore, an interesting class of Riemannian submersion π: (N, h) → (M, g) is the one of homogeneous fibre bundles, of which the twistor bundles Zp(M) are particular examples (p ∈ N∗ ). For example, vertically harmonic sections of homo- geneous fibre bundles have been investigated by C.M. Wood [71, 72].
- 28. INTRODUCTION 13 0.5. Related subjects and works, and motivations. 0.5.1. Relations with surface theory. The theory of harmonic maps of surfaces has been greatly enriched by ideas and methods from integrable systems [12, 16, 14, 18, 23, 31, 32, 67]. In particular, these ideas have revolutionised the theory of harmonic maps from a surface into a symmetric space and so, via an appropriate Gauss map, the theory of constant mean curvature surfaces and Willmore surfaces among others. For example, Pinkall and Sterling [59] were able to give an algebro- geometric construction of all constant mean curvature tori while Dorfmeister-Pedit- Wu [23] gave a Weierstrass formula for all constant mean curvature immersions of any (simply connected) surface in terms of holomorphic data. These advances were taken up by Hélein and Romon [33, 34, 35] who showed that similar ideas could be applied to the study of Hamiltonian stationary Lagrangian surfaces in a 4- dimensional Hermitian symmetric space. It was the first example of second elliptic integrable system associated to a 4-symmetric space. In [42], we presented a new class of isotropic surfaces in the Euclidean space of dimension 8 by identifying R8 with the set of octonions O, and we proved that these surfaces are solutions of a second elliptic integrable system associated to a 4-symmetric space. By restriction to R4 = H, we obtained the Hamiltonian stationary Lagrangian surfaces and by restriction to R3 = Im H, we obtained the CMC surfaces. Furthermore, in [44], we presented a geometric interpretation of all the second elliptic integrable systems associated to a 4-symmetric space in terms of vertically harmonic twistor lifts of conformal immersions into the Riemannian symmetric space (associated to our 4- symmetric space) (see also [17]). When the previous Riemannian symmetric space is 4-dimensional, then any conformal immersion admits an unique twistor lift and the vertical harmonicity of this twistor lift is equivalent to the holomorphicity of the mean curvature vector of the conformal immersion (see [17]). In particular, when the Riemannian symmetric space is Hermitian, one obtains a conceptual explanation of the result of Hélein-Romon. 0.5.2. Relations with mathematical physics. Metric connections with totally skew-symmetric torsion. We refer the reader to [2, 24] about connections with skew-symmetric torsions and their relations to physics and more particulary string theory. Linear metric connections with totally skew-symmetric torsion recently became a subject of interest in theoretical and mathematical physics. Let us give here some examples (taken from [24]). • The target space of supersymmetric sigma models with Wess-Zumino term carries a geometry of a metric connection with skew-symmetric torsion. • In supergravity theories, the geometry of the moduli space of a class of black holes is carried out by a metric connection with skew-symmetric torsion. • The geometry of NS-5 brane solutions of type II supergravity theories is generated by a metric connection with skew-symmetric torsion. • The existence of parallel spinors with respect to a metric connection with skew-symmetric torsion on a Riemannian spin manifold is of importance in string theory, since they are associated with some string solitons (BPS solitons). The sigma-models. The classical non-linear sigma model describes harmonic maps between two (pseudo)riemannian manifolds, which are called the spacetime and the target space. Nonlinear sigma-models provide a much-studied class of field theories of both phenomenological and theoretical interest. The chiral model for
- 29. 14 IDRISSE KHEMAR example summarizes many low energy QCD interactions while 2-dimensional sigma- models may possess nontrivial classical field configurations and have analogies with 4-dimensional Yang-Mills equations but are simpler to handle. The study of Wess- Zumino (WZ) terms has received considerable attention since they were introduced in four-dimensional chiral field theories as effective Lagrangians describing the low energy consequences of the anomalous Ward identities of the theory . Sigma-models also have connections with string theories. Two dimensional sigma-models have already proven a fertile arena for the inter- play of topology, geometry, and physics. The supersymmetric sigma-models have been generalized by introducing a Wess-Zumino term into the Lagrangian. This term may be interpreted as adding torsion to the canonical Levi-Civita connection of the earlier models. The addition of such torsion can impose constraints on the possible geometries of the target [13]. Our contribution. We give new examples of integrable two-dimensional non- linear sigma models. These new examples take place in some homogeneous spaces, namely k -symmetric spaces, which are not symmetric spaces. At our knowledge, all the already known integrable two-dimensional nonlinear sigma models take place in symmetric spaces or (equivalently) in Lie groups. 0.5.3. Relations of F-stringy harmonicity and supersymmetry. The P.D.E of F-stringy harmonicity splits following the splitting TN = H ⊕ V defined by the f-structure F. More precisely, this equation is a coupling between the equation of ¯ J-stringy harmonicity and the equation of vertical harmonicity, the coupling terms calling out the curvatures of H and V, and the component NF |V×H of the Nijenhuis tensor. Moreover, let us suppose that we have we have a fibration π: N → M. Then we have a supersymmetric interpretation of the F-stringy harmonicity: F-stringy harmonicity can be viewed as a supersymmetric extension of the J-stringy har- monicity. In the splitting TN = H ⊕ V, the horizontal subbundle played the role of the odd part and the vertical subbundle plays the role of the even part. In other words, the bosonic equation is a harmonic map equation (the vertical harmonicity) and the fermionic equation is the ¯ J-stringy harmonic map equation. Let us also mention that, in [43], we obtained a supersymmetric interpretation of all the second elliptic integrable systems associated to a 4-symmetric space in terms of superharmonic maps. Acknowledgements. The author wishes to thank Josef Dorfmeister for his useful comments on the first parts of this paper. He is also grateful to him for his interest in the present work, his encouragements as well as his support throughout the preparation of this paper. This work started at T.U. Munich and was completed at Universität Tübingen. The author would like to thank these two institutions. He also wishes to thank Franz Pedit for allowing him to finish this work in very good conditions and an excellent environment at the Mathematisches Institut of Tübingen.
- 30. Notation, conventions and general definitions 0.6. List of notational conventions and organisation of the paper. • Let k ∈ N∗ . Then we will often confuse - when it is convenient to do it- an element in Zk with one of its representants. For example, let (ai)i∈Zk be a family of elements in some vector space E, and 0 ≤ m k/2 an integer. Then we will write ai = a−i 1 ≤ i ≤ m to say that this equality holds for all i ∈ {1 + kZ, . . . , m + kZ} ⊂ Zk. • Let us suppose that a vector space E admits some decomposition E = ⊕i∈I Ei. Then, for any vector v ∈ E we denote by [v]Ei its component in Ei. It could also happen sometimes that we write this component [v]Ei . • We denote EC := E ⊗ C the complexification of a real vector space. • Furthermore let A ∈ End(E) be an endomorphism of a finite dimensional real vector space that we suppose to be diagonalizable. Then if its complex spectrum is {λi, i ∈ I}, we will write its eigenspace decomposition in the form EC = ⊕i∈I EC i , where EC i = ker(A − λiId). Moreover, if λi ∈ R for some i ∈ I, then we set Ei := EC i ∩ E so that EC i = (Ei)C , for these particular i ∈ I. • We denote by ωp a p-th primitive root of unity, which will be often chosen equal to e2iπ/p . • We denote by C∞ (M, N) the set of smooth maps from a manifold M into a manifold N. Now, let π: N → M be a surjection, then we denote by C(π) the set of sections of π, i.e. the maps s such that π ◦ s = IdM . If there is no risk of confusion we will also use the notation C(N). Furthermore, let p: E → M be a vector bundle. Then we will denote by the same letter p its tensorial extensions: p: End(E) → M and so on. • Let (M, g) be a Riemannian manifold. Then we denote by ∗ its Hodge opera- tor (and ∗g if the metric need to be precised). Let (E, h) → M be a Riemannian vec- tor bundle over a manifold M, we denote by Σ(E) the bundle of orthogonal almost complex structures in E. In particular if E = TM then we set Σ(M) := Σ(TM). • More generally, let Z(Rn ) ⊂ End(Rn ) (resp. O(n)) be some submanifold defined for any n ∈ N∗ . This allows us to define Z(E) for any (Riemannian) vector bundle E and we set in particular Z(M) = Z(TM) for any (Riemannian) manifold. • Moreover it will happen that we will write ”SO(E)” without precising that the Riemannian vector bundle E is supposed to be oriented. We will consider that this hypothesis is implicit once we write this kind of symbol. • Let (A, +, ×) be an associative K-algebra over the field K. Then for any a ∈ A, we set Com(a) = {b ∈ A|ab = ba} and Ant(a) = {b ∈ A|ab = −ba}. • w.r.t. : with respect to. A list of notation and an index are available at the end of the paper. Bibli- ographic remarks and a summary of our own contributions are 15
- 31. 16 NOTATION, CONVENTIONS AND GENERAL DEFINITIONS available at the end of each chapter from chapter 2 to section 7. The numbering of the equations is made by section: e.g. equation (2.13) is the 13th equation of chapter 2. Moreover the numbering of theorems, definitions and such is made by subsection: e.g. Theorem 2.2.1 is the first theorem in subsection 2.2 (but it is not in subsubsection 2.2.1). 0.7. Almost complex geometry. Let E be a real vector space endowed with a complex structure: J ∈ End(E), J2 = −Id. Then we denote by E1,0 and E0,1 respectively the eigenspaces of J associated to the eigenvalues ±i respectively. Then we have the following eigenspace decomposition (0.3) EC = E1,0 ⊕ E0,1 and the following equalities (0.4) E1,0 = ker(J − iId) = (J + iId)EC E0,1 = ker(J + iId) = (J − iId)EC so that remarking that (J ± iId)iE = (Id ∓ iJ)E = (Id ∓ iJ)JE = (J ± iId)E, we can also write (0.5) E1,0 = (J + iId)E = (Id − iJ)E = {X − iJX, X ∈ E} E0,1 = (J − iId)E = (Id + iJ)E = {X + iJX, X ∈ E} In the same way we denote by (E∗ )C = E∗ 1,0 ⊕ E∗ 0,1 the decomposition induced on the dual E∗ by the complex structure J∗ : η ∈ E∗ → ηJ ∈ E∗ . Besides, given a vector Z ∈ EC , we denote by Z = [Z]1,0 + [Z]0,1 its decomposition according to (0.3). Let us remark that [Z]1,0 = (Id − iJ)Z and [Z]0,1 = (Id + iJ)Z. Moreover, given η a n-form on E, we denote by η(p,q) its component in Λp,q E∗ according to the decomposition Λn E∗ = ⊕p+q=nΛp,q E∗ , where Λp,q E∗ = Λp E∗ 1,0 ∧ Λq E∗ 0,1 . However for 1-forms, we will often prefer the notation η = η + η , where η and η denote respectively η(1,0) and η(0,1) . More generally, all what precedes holds naturally when E is a real vector bundle over a manifold M, endowed with a complex structure J. We will write d = ∂ + ¯ ∂ the decomposition of the exterior derivative of differential forms on an almost com- plex manifold (M, J), according to the decomposition TMC = T1,0 M ⊕ T0,1 M. We will denote by Hol((M, JM ), (N, JN )) := {f ∈ C∞ (M, N)| df ◦ JM = JN ◦ df} the set of holomorphic maps between two almost complex manifolds (M, JM ) and (N, JN ). In this paper, we will use the following definitions.
- 32. NOTATION, CONVENTIONS AND GENERAL DEFINITIONS 17 Definition 0.7.1. Let E be a real vector bundle. An f-structure in E is an endomorphism F ∈ C(EndE) such that F3 + F = 0. An f-structure on a manifold M is an f-structure in TM. A manifold (M, F) endowed with an f-structure is called an f-manifold. An f-structure F in a vector bundle E is determined by its eigenspaces decom- position that we will denote by EC = E+ ⊕ E− ⊕ E0 where E± = ker(F ∓ iId) and E0 = ker F. In particular if E = TM, then we will set Ti M = (TM)i , ∀i ∈ {0, ±1}. Definition 0.7.2. Let (M, FM ) and (N, FN ) be f-manifolds. Then a map f : (M, FM ) → (N, FN ) is said to be f-holomorphic if it satisfies df ◦ FM = FN ◦ df Definition 0.7.3. Let (M, JM ) be an almost complex manifold and N a man- ifold with a splliting TN = H ⊕ V. Let us suppose that the subbundle H is naturally endowed with a complex structure JH . Then we will say that a map f : (M, JM ) → N is H-holomorphic if it satisfies the equation [df]H ◦ JM = JH ◦ [df]H , where [df]H is the projection of df on H along V. Moreover, if for some reason, H inherits the name of horizontal subbundle, then we will say that f is horizontally holomorphic. Remark 0.7.1. Let us remark that an f-structure in a manifold N is equivalent to a splitting TN = V ⊕ H together with a complex structure JH on H. Definition 0.7.4. An affine manifold (N, ∇) is a manifold endowed with a linear connection. An almost complex affine manifold (N, J, ∇) is an almost complex manifold endowed with an almost complex connection: ∇J = 0.
- 34. CHAPTER 1 Invariant connections on reductive homogeneous spaces The references for this chapter where we recall some results that we will need in this paper, are [54], [61], [20], and to a lesser extent [1] and [37]. 1.1. Linear isotropy representation. Let M = G/H be a homogeneous space with G a real Lie group and H a closed subgroup of G. G acts transitively on M in a natural manner which defines a natural representation: φ: g ∈ G → (φg : x ∈ M → g.x) ∈ Diff(M). Then ker φ is the maximal normal subgroup of G contained in H. Further, let us consider the linear isotropy representation: ρ: h ∈ H → dφh(x0) ∈ GL(Tx0 M) where x0 = 1.H is the reference point in M. Then we have ker ρ ⊃ ker φ. Moreover the linear isotropy representation is faithful (i.e. ρ is injective) if and only if G acts freely on the bundle of linear frames L(M). We can always suppose without loss of generality that the action of G on M is effective (i.e. ker φ = {1}) but it does not imply in general that the linear isotropy representation is faithful. However if there exists on M a G-invariant linear connection, then the linear isotropy representation is faithful provided that G acts effectively on M. (Indeed, given a manifold M with a linear connection, and x ∈ M, an affine transformation f of M is determined by (f(x), df(x)), i.e. f is the identity if and only if it leaves one linear frame fixed.) 1.2. Reductive homogeneous space. Let us suppose now that G/H is reductive, i.e. there exists a decomposition g = h⊕m such that m is AdH-invariant: ∀h ∈ H, Adh(m) = m. Then the surjective map ξ ∈ g → ξ.x0 ∈ Tx0 M has h as kernel and so its restriction to m is an isomorphism m ∼ = Tx0 M. This provides an isomorphism of the associated bundle G ×H m with TM by: (1.1) [g, ξ] → g.(ξ.x0) = Adg(ξ).x where x = π(g) = g.x0. Moreover, we have a natural inclusion G ×H m → G ×H g and the associated bundle G ×H g is canonically identified with the trivial bundle M × g via (1.2) [g, ξ] → (π(g), Adg(ξ)). Thus we have an identification of TM with a subbundle [m] of M × g, which we may view as a g-valued 1-form β on M given by: βx(ξ.x) = Adg[Adg−1 (ξ)]m, 19
- 35. 20 IDRISSE KHEMAR where π(g) = x, ξ ∈ g and [ ]m is the projection on m along h. Equivalently, for all X ∈ TxM, β(X) is the unique element ξ ∈ [m]x (= Adg(m), with π(g) = x) such that X = ξ.x, in other words β(X) is characterized by β(X) ∈ [m]x ⊂ g and X = β(X).x . In fact, β is nothing but the projection on M of the H-equivariant 1-form, θm, on G, i.e. θm is the H-equivariant lift of β. Here, θm is defined as the m-component of the left invariant Maurer-Cartan form θ of G. This can be written as follows (1.3) (π∗ β)g = Adg(θm) ∀g ∈ G with θg(ξg) = g−1 .ξg for all g ∈ G, ξg ∈ TgG. Notation. For any AdH-invariant subspace l ⊂ m, we will denote by [l] the subundle of [m] ⊂ M × g defined by [l]g.x0 = Adg(l). 1.3. The (canonical) invariant connection. On a reductive homogeneous space M = G/H, the Ad(H)-invariant summand m provides by left translation in G, a G-invariant distribution H(m), given by H(m)g = g.m which is horizontal for π: G → M and right H-invariant and thus defines a G-invariant connection in the principal bundle π: G → M. In fact this procedure defines a bijective correspondance between reductive summands m and G-invariant connections in π: G → M (see [54], chap. 2, Th 11.1). Then the corresponding h-valued connection 1-form ω on G (of this G-invariant connection) is the h-component of the left invariant Maurer-Cartan form of G: ω = θh. 1.4. Associated covariant derivative. The connection 1-form ω induces a covariant derivative in the associated bundle G ×H m ∼ = TM and thus a G-invariant covariant derivative ∇0 in the tangent bundle TM. In particular, we can conclude according to section 1.1 that if G/H is reductive then the linear isotropy representation is faithful (provided that G acts effectively) or equivalently that ker Adm = ker ρ = ker φ. We will suppose in the following that, without explicit reference to the contrary, the action of G is effective and (thus) the linear isotropy representation is faithful. One can compute explicitly ∇0 . Lemma 1.4.1. [20] β(∇0 XY ) = X.β(Y ) − [β(X), β(Y )], X, Y ∈ Γ(TM). Let us write (locally) β(X) = AdU(Xm), β(Y ) = AdU(Ym) where U is a (local) section of π and Xm, Ym ∈ C∞ (M, m) then we have (using the previous lemma) β(∇0 XY ) = AdU (dYm(X) + [α(X), Ym] − [Xm, Ym]) = AdU (dYm(X) + [αh(X), Ym] + [αm(X) − Xm, Ym]) where α = U−1 .dU. Besides since U is a section of π (π ◦ U = Id), then pulling back (1.3) by U, we obtain β = AdU(αm) and then αm(X) = Xm, so that (1.4) β(∇0 XY ) = AdU (dYm(X) + [αh(X), Ym]) Remark 1.4.1. We could also say that Xm, Ym are respectively the pullback by U of the H-equivariant lifts X̃, Ỹ of X, Y (given by β(Xπ(g)) = Adg(X̃(g))).
- 36. 1. INVARIANT CONNECTIONS ON REDUCTIVE HOMOGENEOUS SPACES 21 Then ∇0 X Y lifts as the m-valued H-equivariant map on G: ∇0 XY = dỸ (X̃) + [θh(X̃), Ỹ ] and then taking the U-pullback we obtain the previous result (without using lemma 1.4.1). Moreover, we can express ∇0 in terms of the flat differentiation in the trivial bundle M × g (⊃ [m]). Let us differentiate the equation Y = AdU(Ym) (we use the identification TM = [m] ⊂ M × g) dY =AdU (dYm +[α, Ym])=AdU (dYm + [αh, Ym])+AdU (([αm, Ym]))=∇0 Y+[β, Y ]. Finally, we obtain (1.5) dY = ∇0 Y + [β, Y ] and we recover lemma 1.4.1. 1.5. G-invariant linear connections in terms of equivariant bilinear maps. Now let us recall the following results about invariant connections on reductive homogeneous spaces. Theorem 1.5.1. [54] Let πP : P → M be a K-principal bundle over the re- ductive homogeneous space M = G/H and suppose that G acts on P as a group of automorphisms and let u0 ∈ P be a fixed point in the fibre of x0 ∈ M (πP (u0) = x0). There is a bijective correspondance between the set of G-invariant connections ω in P and the set of linear maps Λm : m → k such that (1.6) Λm(hXh−1 ) = λ(h)Λm(X)λ(h)−1 for X ∈ m and h ∈ H where λ: H → K is the morphism defined by hu0 = u0λ(h) (H stabilizes the fibre Px0 = u0.K). The correspondance is given by (1.7) Λ(X) = ωu0 (X̃), ∀X ∈ g where X̃ is the vector field on P induced by X (i.e. ∀u∈P, X̃(u)= d dt |t=0 exp(tX).u) and Λ: g → k is defined by Λ|m = Λm and Λ|h = λ (hence completely determined by Λm). Corollary 1.5.1. In the previous theorem, let us suppose that P is a K- structure on M = G/H, i.e. P is a subbundle of the bundle L(M) of linear frames on M with structure group K ⊂ GL(n, R) = GL(m) (we identify as usual m with Tx0 M by ξ → ξ.x0, and Tx0 M to Rn via the linear frame u0 ∈ P ⊂ L(M)). Then in terms of the G-invariant covariant derivative ∇ corresponding to ω, the G-invariant linear connection in P, the previous bijective correspondance may be given by Λ(X)(Y ) = ∇X̃Ỹ where X̃, Ỹ are any (local) left G-invariant vector fields extending X, Y i.e. there exists a local section, g: U ⊂ M → G, of π: G → M, such that X̃x = Adg(x)(X).p, for all x ∈ M. Remark 1.5.1. In theorem 1.5.1, the G-invariant connection in P defined by Λm = 0 is called the canonical connection (with respect to the decomposition g = h + m). If we set P(M, K) = G(G/H, H) with group of automorphisms
- 37. 22 IDRISSE KHEMAR G, the G-invariant connection defined by the horizontal distribution H(m) is the canonical connection. Now, let P be a G-invariant K-structure on M = G/H as in corollary 1.5.1. Let P be a G-invariant subbundle of P with structure group K ⊂ K, then the canonical connection in P defined by Λm = 0 is (the restriction of ) the canonical connection in P which is itself the restriction to P of the canonical connection in L(M). In particular, if we set P = G.u0, this is a subbundle of P with group H, which is isomorphic to the bundle G(G/H, H). Then the canonical linear connec- tion in P corresponds to the invariant connection in G(G/H, H) defined by the distribution H(m). Theorem 1.5.2. Let P ⊂ L(M) be a K-structure on M = G/H. Then the canonical linear connection (Λm = 0) in P defines the covariant derivative ∇0 in TM (obtained from H(m) in the associated bundle G×H m ∼ = TM). Moreover there is a bijective correspondence between the set of of G-invariant linear connections ∇, on M, determined by a connection in P, and the set of linear maps Λm : m → k ⊂ gl(m) such that (1.8) Λm(hXh−1 ) = Adm(h)Λm(X)Adm(h)−1 ∀X ∈ m, ∀h ∈ H, given by ∇ = ∇0 + Λ̄m i.e. ∇X Y = ∇0 XY + Λ̄m(X)Y for any vector fields X, Y on M, where with the help of ( 1.8) we extended the Ad(H)-equivariant map Λm : m × m → m to the bundle G ×H m = TM to obtain a map Λ̄m : TM × TM → TM. Example 1.5.1. Let us suppose that M is Riemannian (i.e. AdmH is compact and m is endowed with an AdH invariant inner product which defines a G-invariant metric on M) and let us take P = O(M) the bundle of orthonormal frames on M, the previous correspondance is between the set of G-invariant metric linear connections and the set of Ad(H)-equivariant linear maps Λm : m → so(m). In particular the canonical connection ∇0 is metric (for any G-invariant metric on M). Theorem 1.5.3. •: G-invariant tensors on the reductive homogeneous space M = G/H (or more generally G-invariant sections of associated bundles) are parallel with respect to the canonical connection. •: The canonical connection is complete (the geodesics are exactly the curves xt = exp(tX).x0, for X ∈ m). •: Let P be a G-invariant K-structure on M = G/H, then the G-invariant connection defined by Λ: m → k has the same geodesics as the canonical connection if and only if Λm(X)X = 0, ∀X ∈ m Theorem 1.5.4. The torsion tensor T and the curvature tensor R of the G- invariant connection corresponding to Λm is given at the origin point x0 as follows: (1) T(X, Y ) = Λm(X)Y − Λm(Y )X − [X, Y ]m, (2) R(X, Y ) = [Λm(X), Λm(Y )] − Λm([X, Y ]m) − adm([X, Y ]h), for X, Y ∈ m. In particular, for the canonical connection we have T(X, Y ) = −[X, Y ]m and R(X, Y ) = −adm([X, Y ]h), for X, Y ∈ m; moreover we have ∇T = 0, ∇R = 0.
- 38. 1. INVARIANT CONNECTIONS ON REDUCTIVE HOMOGENEOUS SPACES 23 1.6. A family of connections on the reductive space M. We take in what precedes (i.e. in section 1.5) P = L(M). Then let us consider the one parameter family of connections ∇t , 0 ≤ t ≤ 1 defined by Λt m(X)Y = t[X, Y ]m, 0 ≤ t ≤ 1. For t = 0, we obtain the canonical connnection ∇0 . Since for any t ∈ [0, 1], Λt m(X)X = 0, ∀X ∈ m, ∇t has the same geodesics as ∇0 and in particular is complete. The torsion tensor is given (at x0) by (1.9) Tt (X, Y ) = (2t − 1)[X, Y ]m. In particular ∇ 1 2 is the unique torsion free G-invariant linear connection having the same geodesics as the canonical connection (according to theorems 1.5.3 and 1.5.4). Assume now that M is Riemannian, and let us take P = O(M). Then ∇t is metric if and only if Λt m takes values in k = so(m) which is equivalent (for t = 0) to say that M is naturally reductive (which means by definition that ∀X ∈ m, [X, ·]m is skew symmetric). Now (still in the Riemannian case) let us construct a family of linear connections, met ∇t , 0 ≤ t ≤ 1, which are always metric: met ∇t = ∇0 + t [ , ][m] + UM where UM : TM ⊕ TM → TM is the ”natural reductivity term” which is the symmetric bilinear map defined by1 (1.10) UM (X, Y ), Z = [Z, X][m], Y + X, [Z, Y ][m] for all X, Y, Z ∈ [m]. Since UM is symmetric, the torsion of met ∇t is once again given by Tt (X, Y ) = (2t − 1)[X, Y ][m] and thus met ∇ 1 2 is torsion free and metric and we recover that met ∇ 1 2 is the Levi-Civita connection met ∇ 1 2 = ∇L.C. . Obviously if M is naturally reductive then met ∇t = ∇t , ∀t ∈ [0, 1]. Moreover if M is (locally) symmetric, i.e. [m, m] ⊂ h, then all these connections coincide and are equal to the Levi-Civita connection: met ∇t = ∇t = ∇0 = ∇L.C. . Remark 1.6.1. ∇1 is interesting since it is nothing but the flat differentiation in the trivial bundle M × g followed by the projection onto [m] (along [h]) (see remark 1.4.1). So this connection is very natural and following [1], we will call it the anticanonical connection. 1UM is the G-invariant extension of Um : m ⊕ m → m, its restriction to m ⊕ m.
- 39. 24 IDRISSE KHEMAR 1.7. Differentiation in End(T(G/H)). According to section 1.2, we have End(T(G/H)) = G ×H End(m) ⊂ (G/H) × End(g), the previous inclusion being given by [g, A] → (π(g), Adg ◦ A ◦ Adg−1 ) and we embed End(m) in g by extending any endomorphism of m to the corresponding endomorphism of g which vanishes on h. In other words End(T(G/H)) can be iden- tified to the subbundle [End(m)] of the trivial bundle (G/H) × End(g), with fibers [End(m)]g.x0 = End(Adg(m)) = Adg(End(m))Adg−1 = Adg(End(m) ⊕ {0})Adg−1 . Now, let us compute in terms of the Lie algebra setting, the derivative of the inclusion map I: End(T(G/H)) → M × End(g) or more concretely the flat derivative in M × End(g) of any section of End(T(G/H)). To do that, we compute the derivative of Ĩ: (g, Am) ∈ G × End(m) −→ (g.x0, Adg ◦ Am ◦ Adg−1 ) ∈ M × End(g). We obtain dĨ(g, Am) = Adg(θm) . π(g), Adg (dAm + [adθ, Am]) Adg−1 . Next, let us decompose the endomorphisms in g into blocs according to the vector space decomposition g = h ⊕ m: (1.11) End(g) = End(h) End(m, h) End(h, m) End(m) . By regrouping terms, we obtain the following splitting End(g) = End(m) ⊕ (End(m, h) ⊕ End(h, m) ⊕ End(h)) , which applied to dĨ(g, Am), gives us the decomposition dĨ(g, Am) = 0, Adg (dAm + [admθh, Am] + [[admθm]m, Am]) Adg−1 + Adg(θm) . π(g), Adg ([admθm]h ◦ Am − Am ◦ adhθm) Adg−1 . (1.12) The first term is in the vertical space VĨ(g,Am) =Adg(End(m))Adg−1 =End(Tπ(g)M) and the previous decomposition (1.12) provides us with a splitting TEnd(M) = V ⊕ H = π∗ M (End(M)) ⊕ H, i.e. a connection on End(M). Let us determine this connection: we see that the projection on the vertical space (along the horizontal space) corresponds to the projection on [End(m)] following (1.11) so that according to remark 1.6.1, we can conclude that the horizontal distribution H defines the connection ∇1 on End(TM) = TM∗ ⊗ TM. Remark 1.7.1. a) We can recover this previous fact directly from the first term of (1.12) and the definition of ∇1 . Indeed, first recall that given two linear connections ∇, ∇ on M, we can write ∇ = ∇ + F, where F is a section of TM∗ ⊗ End(TM), and then for any section A in End(TM), ∇ A = ∇A + [F, A]. Besides, ∇1 = ∇0 + [ , ][m], and moreover, if we write (locally) A = (π(U), AdU ◦ Am ◦ AdU−1 ) where U is a local section of π and Am ∈ C∞ (M, End(m)), then according to (1.4), (1.13) ∇0 A = AdU (dAm + [admαh, Am]) ,
- 40. 1. INVARIANT CONNECTIONS ON REDUCTIVE HOMOGENEOUS SPACES 25 so that we conclude that ∇1 A = AdU (dAm + [admαh, Am] + [[admαm]m, Am]) AdU−1 which is the (pullback of) the first term of (1.12). b) Furthermore, if G/H is (locally) symmetric (i.e. [m, m] ⊂ h), then ∇L.C. = ∇0 = ∇1 and in particular (1.14) ∇L.C. A = AdU (dAm + [admαh, Am]) .
- 42. CHAPTER 2 m-th elliptic integrable system associated to a k -symmetric space 2.0.1. Definition of Gτ (even when τ does not integrate in G). Here, we will extend the notion of subgroup fixed by an automorphism of Lie group to the sit- uation where only a Lie algebra automorphism is provided. Indeed, let τ : G → G be a Lie group automorphism, then usually one can define Gτ = {g ∈ G| τ(g) = g} the subgroup fixed by τ. Now, we want to extend this definition to the situation where we only have a Lie algebra automorphism, and so that the two definitions coincide when the Lie algebra automorphism integrates in G. Let g be a real Lie algebra and τ : g → g be an automorphism. Then let us denote by (2.1) g0 = gτ := {ξ ∈ g| τ(ξ) = ξ} the subalgebra of g fixed by τ. Let us assume that τ defines in g a τ-invariant reductive decomposition g = g0 ⊕ n, [g0, n] ⊂ n, τ(n) = n. Moreover we suppose that we have n = Im (Id − τ), i.e. that the decomposition g = g0 ⊕ n coincides with the Fitting decomposition of Id − τ (remark that this decomposition is then automatically reductive since adξ ◦ (Id − τ) = (Id − τ) ◦ adξ, ∀ξ ∈ g0). Without loss of generality, we assume that the center of g is trivial. Moreover, we assume also, without loss of generality, that g0 does not contain a nontrivial ideal of g, i.e. that adn : g0 → gl(n) is injective (the kernel is a τ-invariant ideal of g that we factor out). We then have (2.2) g0 = {ξ ∈ g| τ ◦ adξ ◦ τ−1 = adξ} Let G be a connected Lie group with Lie algebra g. Then let us consider the subgroup G0 = {g ∈ G| τ ◦ Adg ◦ τ−1 = Adg}. Then G0 is a closed subgroup of G and Lie G0 = g0. Moreover, without loss of generality, we will suppose that G0 does not contain non-trivial normal subgroup of G (by factoring out, if needed, by some discrete subgroup of G), i.e. that Adn : G0 → GL(n) is injective (see section 1). Now, we want to prove that if τ integrates in G, then we have G0 = Gτ , where Gτ is the subgroup fixed by τ : G → G. First, we have ∀g ∈ Gτ , τ ◦ Adg ◦ τ−1 = Adτ(g) = Adg, thus Gτ ⊂ G0. Conversely, we have ∀g ∈ G0, Adg(n) = n and Adg = τ ◦ Adg ◦ τ−1 = Adτ(g) so that Adng = Adnτ(g) and thus g = τ(g) since Adn : G0 → GL(n) is injective. We have proved Gτ = G0. This allows us to make the following: 27
- 43. 28 IDRISSE KHEMAR Definition 2.0.1. Let g be a real Lie algebra and τ : g → g be an automor- phism, and G a connected Lie group with Lie algebra g. Let us assume that τ defines in g a τ-invariant reductive decomposition: g = g0 ⊕ n with n = Im (Id − τ). Then we will set Gτ := {g ∈ G| τ ◦ Adg ◦ τ−1 = Adg}. Let us conclude this subsection by some notation: Notation and conventions. In all the paper, when a real Lie algebra g and an automorphism τ will be given, then we will suppose without loss of generality that the center of g is trivial, g0 will denote the Lie subalgebra defined by (2.1), G will denote a connected Lie group with Lie algebra g and G0 ⊂ G a closed subgroup such that (Gτ )0 ⊂ G0 ⊂ Gτ (which implies that its Lie algebra is g0). Moreover, without loss of generality, we will always suppose that g0 does not contain non-trivial ideal of g - we will then say that (g, g0) is effective - and also suppose that Gτ does not contain non-trivial normal subgroup of G (by factoring out, if needed, by some discrete subgroup of G). Consequently, when τ can be integrated in G, then Gτ will coincide with the subgroup of G fixed by τ : G → G. Remark 2.0.2. • Let g be a Lie algebra and τ : g → g be an automorphism. Let us consider G = AdG =: Int(g) the adjoint group of g (which does not depend on the choice of the connected group G), and let C = ker Ad be the center of G. Then we can identify the morphism Ad with the covering π: G → G/C and G to G/C. Besides τ always integrates in G into τ defined by τ = Intτ : Adg ∈ G → τ ◦ Adg ◦ τ−1 and we have τ ◦ π = π ◦ τ (See [36, p. 127]). Then, we see that G0 = Ad−1 (Gτ ) = π−1 (Gτ ). • Now, let us come back to the hypothesis of trivialness of the center of g. Let g be a real Lie algebra with center c and τ : g → g an automorphism. Then we have τ(c) = c. Therefore, if τ is of order k , for any m ∈ N∗ , the m-th elliptic integrable system associated to τ splits into two independent systems: a general one on g = g/c and a trivial one on c = Rn : dαλ = 0, ∀λ ∈ C∗ . Then the solutions of this trivial system are the maps fλ : L → c, fλ = m j=−m λj fj, ∀λ ∈ C∗ , such that f−j = fj, j ≥ 0, and f−j is an holomorphic map into the eigenspace cC ∩ gC −j. (Of course αλ = dfλ.) Therefore, the hypothesis of trivialness of the center can be done without loss of generality. 2.1. Finite order Lie algebra automorphisms. Let g be a real Lie algebra and τ : g → g be an automorphism of order k . Let ωk be a k -th primitive root of unity. Then we have the following eigenspace decomposition: gC = j∈Z/kZ gC j , [gC j , gC l ] ⊂ gC j+l where gC j is the ωj k -eigenspace of τ. We then have to distinguish two cases. 2.1.1. The even case: k = 2k. Then we have gC 0 = (g0)C . Moreover let us remark that (2.3) gC j = gC −j, ∀j ∈ Z/k Z. Therefore gC k = gC −k = gC k so that we can set gC k = (gk)C with gk = {ξ ∈ g| τ(ξ) = −ξ}.
- 44. m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k -SYMMETRIC SPACE 29 Moreover, owing to (2.3), we can define mj as the unique real subspace in g such that its complexified is given by mC j = gC j ⊕ gC −j for j = 0, k, and n as the unique real subspace such that nC = j∈Z k{0} gC j , that is n = (⊕k−1 j=1 mj) ⊕ gk. In particular τ defines a τ-invariant reductive decom- position g = g0 ⊕ n. Hence the eigenspace decomposition is written: gC = gC −(k−1) ⊕ . . . ⊕ gC −1 ⊕ gC 0 ⊕ gC 1 ⊕ . . . ⊕ gC k−1 ⊕ gC k so that by grouping gC = gC 0 ⊕ gC k ⊕ ⊕k−1 j=1 mC j = hC ⊕ mC where h = g0 ⊕ gk and m = ⊕k−1 j=1 mj. Considering the automorphism σ = τ2 , we have h = gσ and g = h ⊕ m is the reductive decomposition defined by the order k automorphism σ. Without loss of generality, and according to our convention applied to g and σ, we will suppose in the following that (g, h) is effective i.e. h does not contain a nontrivial ideal of g. This implies in particular that (g, g0) is also effective. Now let us integrate our setting: let G be a Lie group with Lie algebra g and we choose G0 such that (Gτ )0 ⊂ G0 ⊂ Gτ . Then G/G0 is a (locally) 2k- symmetric space (it is globally 2k-symmetric if τ integrates in G) and is in particular a reductive homogeneous space (reductive decomposition g = g0 ⊕ n). Moreover since σ = τ2 is an order k automorphism, then for any subgroup H, such that (Gσ )0 ⊂ H ⊂ Gσ , G/H is a (locally) k-symmetric space. In all the following we will always do this choice for H and suppose that H ⊃ G0 (it is already true up to covering since h ⊃ g0) so that N = G/G0 has a structure of associated bundle over M = G/H with fibre H/G0: G/G0 ∼ = G ×H H/G0. We can add that on h, τ is an involution: (τ|h)2 = Idh, whose symmetric decomposition is h = g0 ⊕ gk, and gives rise to the (locally) symmetric space H/G0. The fibre H/G0 is thus (locally) symmetric (and globally symmetric if the inner automorphism Intτ|m stabilizes AdmH). Owing to the effectivity of (g, h), we have the following characterization: g0 = {ξ ∈ h|[admξ, τ|m] = 0} (2.4) gk = {ξ ∈ h|{admξ, τ|m} = 0} (2.5) {} being the anticommutator. Two different types of 2k-symmetric spaces. Since (τ|m)k is an involution, there exist two invariant subspaces m and m , of m, each sum of certain mj’s, such that (τ|m)k = −Idm ⊕ Idm .
- 45. 30 IDRISSE KHEMAR These subspaces m and m can be computed easily : m = ⊕ [ k−2 2 ] j=0 m2j+1 and m = ⊕ [ k−1 2 ] j=1 m2j. In other words, their complexifications are given by mC = zk = −1 z = −1 ker(τ − zId), mC = zk = 1 z = ±1 ker(τ − zId). At this stage, there are two possibilities: •: if m = 0 then (τ|m)k = −Idm and τ|m admits eigenvalues only in the set {zk = −1, z = −1}. •: if m = 0 then (τ|m)k = −Idm and τ|m admits eigenvalues in each of the sets {zk = 1, z = ±1} and {zk = −1, z = −1}. These two cases give rise to two different types of 2k-symmetric spaces (see sec- tion 3.5) G-invariant metrics. Now, let us suppose that M = G/H is Riemannian (i.e. AdmH is compact) then we can choose an AdH-invariant inner product on m for which τ|m is an isometry1 . In the next of the paper, we will always choose this kind of inner product on m. Therefore, τ|m is an order 2k isometry. We will study this kind of endomorphism in section 3. Moreover, let us remark that if G/H is Riemannian then so is G/G0. Further, since the elliptic system we will study in this paper is given in the Lie algebra setting it is useful to know how the fact that G/H is Riemannian can be read in the Lie algebra setting. In fact, under our hypothesis of effectivity, G/H is Riemannian if and only if h is compactly imbedded2 in g and AdH/AdH0 is finite. Moreover, according to proposition 8.2.3 in the Appendix, AdH/AdH0 is always finite so that G/H is Riemannian if and only if h is compactly imbedded in g. 2.1.2. The odd case: k = 2k + 1. As in the even case we have gC 0 = (g0)C and gC j = gC −j, ∀j ∈ Z/k Z. Then we obtain the following eigenspace decomposition: (2.6) gC = gC −k ⊕ . . . ⊕ gC −1 ⊕ gC 0 ⊕ gC 1 ⊕ . . . ⊕ gC k , which provides in particular the following reductive decomposition: g = g0 ⊕ m with m = ⊕k j=1mj and mj is the real subspace whose the complexification is mC j = gC −j ⊕ gC j . According to our convention, we suppose that (g, g0) is effective. Then, as in the even case, integrating our setting and choosing G0 such that (Gτ )0 ⊂ G0 ⊂ Gτ , we consider N = G/G0 which is a locally (2k + 1)-symmetric space and in particular a reductive homogeneous space. Moreover, the decomposi- tion (2.6) gives rises to a splitting TNC = T1,0 N ⊕ T0,1 N defined by (2.7) TNC = ⊕k j=1[gC −j] ⊕ ⊕k j=1[gC j ] = T1,0 N ⊕ T0,1 N This splitting defines a canonical almost complex structure on G/G0, that we will denote by J. 1See the Appendix, theorem 8.2.2, for the proof of the existence of such an inner product. 2See [36, p. 130] for a definition.
- 46. m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k -SYMMETRIC SPACE 31 Let us suppose that N = G/G0 is Riemannian; then the subgroup generated by AdmG0 and τ|m is compact (because AdmG0 is compact and τ|mAdmg τ−1 |m = Admg, ∀g ∈ G0, and τm is of finite order). Therefore there exists an AdG0-invariant inner product on m for which τ|m is an isometry. In the next of the paper, we will always choose this kind of inner product on m (when N is Riemannian). 2.2. Definitions and general properties of the m-th elliptic system. 2.2.1. Definitions. Let τ : g → g be an order k automorphism with k ∈ N∗ (if k = 1 then τ = Id). We use the notations of 2.1. Let us begin by defining some useful notations. Notation and convention Given I ⊂ N, we denote by j∈I gC j , the product j∈I gC j mod k . In the case j∈I gC j mod k is a direct sum in gC , we will identify it with the previous product via the canonical isomorphism (2.8) (aj)j∈I −→ j∈I aj, and we will denote these two subspaces by the same notation ⊕j∈I gC j . Now, let us define the m-th elliptic integrable system associated to a k - symmetric space, in the sense of Terng [66]. Definition 2.2.1. Let L be a Riemann surface. The m-th (g, τ)-system (with the (−)-convention) on L is the equation for (u0, . . . , um), (1, 0)-type 1-form on L with values in m j=0 gC −j: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ¯ ∂uj + m−j i=0 [ūi ∧ ui+j] = 0 (Sj), if 1 ≤ j ≤ m, ¯ ∂u0 + ∂ū0 + m j=0 [uj ∧ ūj] = 0 (S0) (Syst) It is equivalent to say that the 1-form (2.9) αλ = m j=0 λ−j uj + λj ūj = m j=−m λj α̂j satisfies the zero curvature equation: (2.10) dαλ + 1 2 [αλ ∧ αλ] = 0, ∀λ ∈ C∗ . Definition 2.2.2. Let L be a Riemann surface. The m-th (g, τ)-system (with the (+)-convention) on L is the equation (Syst) as in definition 2.2.1 but for (u0, . . . , um), (1, 0)-type 1-form on L with values in m j=0 gC j :3 It is equivalent to say that the 1-form (2.11) αλ = m j=0 λj uj + λ−j ūj = m j=−m λj α̂j satisfies the zero curvature equation ( 2.10). 3instead of m j=0 gC −j.
- 47. 32 IDRISSE KHEMAR Remark 2.2.1. The difference between the two conventions is that in the first one α λ = m j=0 λ−j uj involves negative powers of λ whereas in the second one α λ involves positive powers of λ (in other words α̂ −j = 0, for j ≥ 1 in the first one whereas α̂ j = 0, for j ≥ 1 in the second one). In fact the second system is the first system associated to τ−1 and vice versa. The first convention is the traditional one: it was used for harmonic maps into symmetric space (see [23]) and by Hélein-Romon [33, 34, 35] for Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric space – first example of second elliptic integrable system associated to a 4-symmetric space. Then this convention was used in [42, 43, 44]. Terng [66], herself, in her definition of the elliptic integrable system uses also this convention. However in [17], it is the second convention which is used. The (+)-convention is in fact the most natural, as we will see, since in the (−)- convention, there is a minus sign which appears when we pass from the Lie algebra setting to the geometric setting. This is what happens for example when we will associate to the automorphism τ, a almost complex complex structure in the target space (see also [44], remark 13). Namely, we will consider that the eigenspace spaces g−j will define the (1, 0)-part and the subspaces gj the (0, 1)-part of the tangent space of the target. But the (+)-convention leads to several changes in the traditional conventions, like for example in the DPW method [23], we must use the Iwasawa decomposition ΛGC τ = ΛGτ .Λ− B GC τ instead of ΛGC τ = ΛGτ .Λ+ B GC τ and in particular the holomorphic potential involves positive power of λ instead of negative one as it is the case traditionally. We decided here to continue to perpetuate the tradition as in [44] and to use the first convention. So in the following when we will speak about the m-th elliptic integrable system, it will be according to the definition 2.2.1. Notation. Sometimes, to avoid confusion we will denote (Syst) either by (Syst(m, g, τ)), (Syst(m, τ)) or simply by (Syst(m)) depending on the context and the needs. For shortness we will also often say the (m, g, τ)-system instead of the m-th (g, τ)-system. We will also say the m-th elliptic (integrable) system associated to (the k -symmetric space) G/G0. We will say that a family of 1-forms (αλ)λ∈C∗ (denoted by abuse of notation, simply by αλ) is solution of the (m, g, τ)-system (or of (Syst)) if it corresponds to some solution u of this system, according to (2.9). Therefore αλ is solution of the (m, g, τ)-system if and only if it can be written in the form (2.9), for some (1, 0)-type 1-form u on L with values in m j=0 gC −j, and satisfies the zero curvature equation (2.10). It will turn out that the (m, g, τ)-system has distinctively different behaviour if k is even and if k is odd. Moreover, for every k there are three different types of behaviour, according to the size of m relatively to k . Definition 2.2.3. If k = 1, set mk = 0. If k 1, we set mk = k + 1 2 = k if k = 2k k + 1 if k = 2k + 1 . We will say that the m-th (g, τ)-system is:
- 48. m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k -SYMMETRIC SPACE 33 – in the primitive case (or that the system is primitive) if 0 ≤ m mk , – in the determined case (or that the system is determined) if mk ≤ m ≤ k − 1, – and in the underdetermined case (or that the system is underdetermined) if m k − 1. Moreover, the determined system of minimal order mk will be called ”the min- imal determined system”, and the one of maximal order k − 1 will be called ”the maximal determined system”. Let us consider the g-valued 1-form α := αλ=1. Then we have α = m j=0 uj +ūj according to (2.9) which is equivalent to α = m j=0 uj, since α is g-valued • In the primitive and determined cases (m ≤ k − 1), m j=0 gC −j is a direct sum so that u = (u0, . . . , um) can be identified with m j=0 uj = α via (2.8) and according to our convention. We will then write simply u = α . In particular we have uj = α −j ∀j, 0 ≤ j ≤ m with αj := [α]gj ∀j ∈ Z/k Z. Hence in the primitive and determined cases the m-th (g, τ)-system can be considered as a system on α. Moreover, we can recover αλ from α and we will speak about the ”extended Maurer Cartan form” αλ which is then associated to α by αλ = m j=1 λ−j α −j + α0 + m j=1 λj α j according to (2.9). • In the underdetermined case, m j=0 gC −j is not a direct sum so that to a given α (coming from some solution αλ of the m-th (g, τ)-system, according to α = αλ=1) there are a priori many (other) corresponding solutions u = (u0, . . . , um) since ∀j ∈ Z/k Z, α −j = i≡j[k] ui. In fact, we will prove that there are effectively an infinity of other solutions αλ satisfying the condition αλ=1 = α (see theorem 2.2.1; see also section 2.5 for a conceptual explanation). 2.2.2. The geometric solution. Convention. Our study, in the present paper, is local therefore we will suppose (when it is necessary to do so) either that L is simply connected or that all lifts (of maps defined on L) and integrations (of 1- forms on L) are made locally. We consider that these considerations are implicit and will not specify these most of the time. The equations (2.10) and (2.9) are invariant by gauge transformations by the group C∞ (L, G0): U0 · αλ = AdU0(αλ) − dU0.U−1 0 . where U0 ∈ C∞ (L, G0). This means that if αλ satisfies (2.10) then so is U0 · αλ and if αλ can be written in the form (2.9) then so is U0 · αλ. Therefore if αλ is a solution of (Syst) then so is U0 ·αλ. This allows us to define a geometric solution of (Syst) as follows: Definition 2.2.4. A map f : L → G/G0 is a geometric solution of (Syst) if for any (local) lift U of f, into G, there exists a (local) solution αλ of (Syst) such that U−1 .dU = αλ=1.
- 49. 34 IDRISSE KHEMAR In other words, we obtain the set of geometric solutions as follows: for each solution αλ of (Syst), consider the g-valued 1-form α := αλ=1, then integrate it by U : L → G, U−1 .dU = α, and finally project U on G/G0 to obtain the map f : L → G/G0. Now, to simplify the exposition, let us suppose that L is simply connected (until the end of 2.2.2). Then (αλ)λ∈C∗ → α = αλ=1 is a surjective map from the set of solutions of (Syst) to the set of Maurer-Cartan forms of lifts of geometric solutions. According to the discussion at the end of subsection 2.2.1, this map is bijective in the primitive and determined case (m ≤ k − 1) and not injective in the underdetermined case (m k − 1). By quotienting by C∞ (L, G0), we obtain a surjective map πm with the same properties, taking values in the set of geometric solutions Let us make more precise all that. We suppose, until the end of this sub- section 2.2.2, that the automorphism τ : g → g is fixed (so that the only data which varies in the (m, g, τ)-system is the order m). First, let us give an explicit expression of the space S(m) of solutions αλ of the system (Syst(m)), i.e. the so- lutions of the zero curvature equation (2.10), which satisfies the equality (2.9) for some (1, 0)-type 1-form u on L with values in m j=0 gC −j. To do that, we want to express the condition to be written in the form (2.9) as a condition on the loop α• : λ ∈ S1 → αλ ∈ C(T∗ L⊗g). The ” • ” means of course functions on the param- eter4 λ ∈ S1 . More precisely, we will consider α• as a 1-form on L with values in the loop Lie algebra C(S1 , g). Then the condition to be written in the form (2.9) means: (2.12) (2.9) ⇐⇒ (α• ∈ Λmgτ and α • ∈ Λ− gC τ ) where Λgτ = {η• : S1 → g| ηωλ = τ(ηλ), ∀λ ∈ S1 } Λmgτ = {η• ∈ Λgτ | ηλ = |j|≤m λj η̂j} Λ− gC τ = {η• ∈ ΛgC τ | ηλ = j≤0 λj η̂j} and ω is a k -th primitive root of unity. We refer the reader to [60] for more details about loop groups ad their Lie algebras (in particular about the possible choices of topology which makes Λgτ be a Banach Lie algebra). Therefore the space of solutions of (Syst(m)) is given by (2.13) S(m) = {α• ∈ C(T∗ L ⊗ Λmgτ )| α • ∈ Λ− gC τ and dα• + 1 2 [α• ∧ α•] = 0}. Let us remark that the condition α • ∈ Λ− gC τ can be interpretated as a condition of C-linearity. Indeed, the Banach vector space Λgτ /g0 is naturally endowed with the complex structure defined by the following decomposition (2.14) (Λgτ /g0)C = ΛgC τ /gC 0 = Λ− ∗ gτ ⊕ Λ+ ∗ gτ , where Λ± ∗ gτ = {η• ∈ ΛgC τ | ηλ = j≷0 λj η̂j}. Then the condition α • ∈ Λ− gC τ means that [α •]∗ : TL → (Λgτ /g0)C is C-linear, where [ ]∗ denotes the component in Λ∗gτ = {η• ∈ Λgτ |ηλ = j=0 λj η̂j} ∼ = Λgτ /g0. 4Remark that α• determines (αλ)λ∈C∗ , when this latter satisfies (2.9).
- 50. m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k -SYMMETRIC SPACE 35 Now let us integrate our setting. Firstly, let us define the twisted loop group ([60]) ΛGτ = {U• : S1 → G|Uωλ = τ (Uλ)}. Then, let us set Em = {U• : L → ΛGτ |Uλ(0) = 1, ∀λ ∈ S1 ; αλ := U−1 λ .dUλ is a solution of (Syst(m))} Em 1 = {U : L → G|∃U• ∈ Em , U = U1} Gm 1 = {f : L → G/G0 geometric solution of (Syst(m)), f(0) = 1.G0} Gm = {f• = πG/G0 ◦ U•, U• ∈ Em } Remark that because of the gauge invariance: E(m).K ⊂ E(m), where K = C∞ ∗ (L, G0) = {U ∈ C∞ (L, G0)|U(0) = 1}, any lift5 U• : L → ΛGτ of an ele- ment f• ∈ Gm belongs to Em . Definition 2.2.5. An element f• ∈ Gm will be called an extended geometric solution of (Syst(m)). The space of geometric solutions is obviously obtained from the space of ex- tended geometric solutions Gm by the evaluation at λ = 1. Moreover S(m) Em is determined by Gm • because of the gauge invariance: E(m).K ⊂ E(m), so that we can write Gm = E(m)/K. Consequently, we have also Gm 1 = Em 1 /K. Finally, we obtain the following diagram S(m) int − − − − → ∼ = Em πK − − − − → Em /K Gm • ev1 ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ πm ⏐ ⏐ S(m)1 int − − − − → ∼ = Em 1 − − − − → Em 1 /K Gm . Therefore, the surjective map πm is bijective for m ≤ k − 1 and not injective for m k − 1 (because so is ev1). We will need the following definition: Definition 2.2.6. Given a g-valued Maurer-Cartan 1-form α on L, we define the geometric map corresponding to α, as f = πG/G0 ◦ U, where U integrates α: U−1 .dU = α, U(0) = 1. We have seen that in the primitive and determined cases, we can consider (Syst(m)) as a system on the g-valued 1-form α := αλ=1. Since the Maurer-Cartan equation for α is always contained in (Syst(m)) according to (2.10), this systems on α is itself equivalent to a system on the geometric map f corresponding to α. This system on f is then a G-invariant elliptic PDE on f of order ≤ 2. Let us summarize: Proposition 2.2.1. The natural surjective map πm : Gm → Gm 1 from the set of extended geometric solutions of the (m, g, τ)-system into the set of geometric solutions is bijective in the primitive and determined cases (m ≤ k − 1) and not injective in the underdetermined case (m k − 1). Moreover, in the primitive and determined cases, the (m, g, τ)-system - which is initially a system on the Λmgτ - valued 1-form αλ - is in fact a system on the 1-form α := αλ=1, itself equivalent to an elliptic PDE of order ≤ 2 on the corresponding geometric map f : L → G/G0. 5With initial condition U(0) = 1.
- 51. 36 IDRISSE KHEMAR Furthermore, let us interpret the C-linearity of [α •]∗ : TL → (Λgτ /g0)C in terms of the corresponding extended geometric solution f• : L → ΛGτ /G0, defined by f• = πG/G0 ◦ U• where U• integrates α•. Firstly, the complex structure defined in Λgτ /g0 by (2.14) is AdG0-invariant so that it defines a ΛGτ -invariant complex structure on the homogeneous space ΛGτ /G0. Therefore the C-linearity of [α •]∗ means exactly that f• : L → ΛGτ /G0 is holomorphic. Now, let us interpret the condition α• ∈ Λmgτ in terms of the map f•. Let us consider the following AdG0- invariant decomposition Λgτ /g0 = Λm∗gτ ⊕ Λmgτ where Λm∗gτ = Λmgτ ∩ Λ∗gτ and Λmgτ = {η• ∈ Λgτ | ηλ = |j|m λj η̂j}, which gives rise respectively to some ΛGτ -invariant splitting T(ΛGτ /G0) = HΛ m ⊕ VΛ m. Then HΛ m and VΛ m inherit respectively the qualificatifs horizontal and vertical sub- bundle respectively. Therefore, in the same spirit as [23] (remark 2.5 and proposi- tion 2.6), the equation (2.13) gives us the following familiar twistorial characteri- zation Proposition 2.2.2. A map f• : L → ΛGτ /G0 is an extended geometric solution of the (m, g, τ)-system if and only if it is holomorphic and horizontal. 2.2.3. The increasing sequence of spaces of solutions: (S(m))m∈N. Again, we suppose in all 2.2.3 that the automorphism τ is fixed and that L is simply connected. Then according to the realisation of (Syst(m)) in the forms (2.10) and (2.9), we see that any solution of (Syst(m)) is solution of (Syst(m )) for m ≤ m (take uj = 0 for m j ≤ m ). More precisely, (Syst(m)) is a reduction of (Syst(m )): (Syst(m)) is obtained from (Syst(m )) by putting uj = 0, m j ≤ m , in (Syst(m )). In particular, S(m) ⊂ S(m ) for m ≤ m ; so that any solution in the primitive case (m mk ) is solution of any determined system (mk ≤ m ≤ k − 1), and any solution of a determined system is solution of any underdetermined system (m k − 1). {Primitive case} ⊂ {determined case} ⊂ {underdetermined case}. Remark 2.2.2. We have πm |Gm = πm if m ≤ m . In particular, πm (Gm ) = Gm 1 . We can set S(∞) = ∪m∈NS(m), E∞ = ∪m∈NEm and G∞ = ∪m∈NGm . Then we have G∞ = E∞ /K. Moreover we can define the surjective map π∞ : G∞ → G∞ 1 such that π∞|Gm = πm, ∀m ∈ N. Then π∞|Gm is a bijection onto Gm 1 for each m ≤ k − 1. We can call S(∞) the (g, τ)-system, and then we can speak about its subsystem of order m, namely S(m). In particular, we have the following characterization: S(∞) = {α• ∈ C(T∗ L ⊗ Λ(∞)gτ )| α • ∈ Λ− gτ and dα• + 1 2 [α• ∧ α•] = 0} where Λ(∞)gτ = ∪m∈NΛmgτ . Important remark. It could happens that the eigenspaces gj vanishes for the first values of j, i.e. j close to 0. For example it is a priori possible that g0 = 0 (which implies that our k -symmetric space is a group). Then for the values of m ≥ k close to k , the underdetermined systems Syst(m) coincide trivially with the determined system Syst(k −1), because then uj ∈ g−j = {0}, for k ≤ j ≤ m. This
- 52. m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k -SYMMETRIC SPACE 37 is why, when we say “underdetermined”, we mean in fact “underdetermined but not determined” to exclude the (potential) formal underdetermined systems which are in fact trivially determined because of the (potential) vanishing of the first eigenspaces. Remark that this eventuality to happen need that all the eigenspaces from 0 to a small value, vanish (they can not all vanish otherwise g = 0), and in particular, this implies that g0 = 0 (the automorphism τ has no fixed point). The primitive and determined cases (m ≤ k − 1). Now, let us apply the previous discussion (about the increasing sequence (S(m))m∈N) to the study of the determined case. Let us recall that in this case, we can consider that the system (Syst(m)) deals only with g-valued 1-forms α. Let us also keep in mind that, in this case, we have α = u ∈ ⊕m j=0gC −j (see the discussion in the end of 2.2.1, after definition 2.2.3). Then we obtain immediately: Proposition 2.2.3. The solutions of a determined system (Syst(m)), mk ≤ m ≤ k − 1, are exactly the solutions of the maximal determined system, i.e. (Syst(k − 1)), which satisfy the holomorphicity conditions: α −j = 0, 1 ≤ j ≤ k − 1 − m. Moreover, the solutions of a primitive system (Syst(m)), 1 ≤ m ≤ mk − 1, are the solutions of the minimal determined system, i.e. (Syst(mk )), which satisfy (i): if k = 2k is even, the horizontality conditions: αk = α±(k−1) = . . . α±(m+1) = 0 (ii): if k = 2k + 1 is odd, • the holomorphicity condition : α −k = 0 if m = k, • the horizontality conditions : α±k =α±(k−1) = . . . α±(m+1) = 0 if m ≤ k − 1. The noninjectivity of πm in the underdetermined case. Now, let us turn ourself to the underdetermined case. We want to study the surjective map πm : Gm → Gm 1 , in this case. Theorem 2.2.1. The surjective map πm : Gm → Gm 1 , from the set of extended geometric solutions into the set of geometric solutions, is a principal bundle with as structure group some group of holomorphic curves into ΛGτ /G0. In the determined and primitive cases, this group is trivial whereas in the underdetermined case it is of infinite dimension. Therefore, in the underdetermined case, the surjective map πm : Gm → Gm 1 is not injective and its fibers are of an infinite dimension (even up to conformal transformations of L). A fortiori, so is for the map ev1 : α• ∈ S(m) → α := αλ=1 ∈ S(m)1. Proof. The natural map ΛGτ → G, g → g(1) is a morphism of group and therefore a principal fibre bundle, with as structure group the kernel H = {g ∈ ΛGτ |g(1) = 1G}. Then it induces the fibration π: ΛGτ /G0 → G/G0 which is also a H-principal bundle (remark that G0 ∩ H = {1}), H acting by the left on ΛGτ /G0. Let Holm (L, ΛGτ /G0) be the set of holomorphic integral curves f• of the holo- morphic ΛGτ -invariant distribution HΛ m, such that f•(o) = 1.G0, where o is a reference point in L. According to proposition 2.2.2, we have Gm = ΛGτ .Holm (L,
- 53. 38 IDRISSE KHEMAR ΛGτ /G0). Therefore, denoting (Gm 1 )o = {g ∈ Gm 1 |g(o) = 1.G0}, we want to prove that the following map (2.15) πm : f• ∈ Holm (L, ΛGτ /G0) → f1 ∈ (Gm 1 )o is a principal fibre bundle. To fix ideas, let us first consider the fibre defined by the constant map 1.G0. This is nothing but the holomorphic integral curves of the holomorphic ΛGτ -invariant distribution HΛ m ∗ defined by the complex subspace {η ∈ Λm∗gτ | η1 = (Jη)1 = 0}, such that f•(o) = 1.G0. Of course J is the complex structure on ΛGτ /G0. Furthermore, remark that HΛ m ∗ vanishes in the determined case and is nontrivial in the undetermined case. More generally, let us compute the fibre of any f• ∈ Holm (L, ΛGτ /G0). Firstly, we remark that H is immersed into ΛGτ /G0 via the projection πG0 : ΛGτ → ΛGτ /G0. The image of H in ΛGτ /G0 is {h ∈ ΛGτ /G0|h(1) = 1.G0}. Remark also that each element of this image has one and only one lift in H ⊂ ΛGτ . Now, let f, f ∈ Holm (L, ΛGτ /G0) be in the same fibre of πm. Therefore, there exists h: L → H such that f = h.f. Moreover it is not difficult to see that h: L → ΛGτ /G0 is holomorphic. Then we have df = dh.f + h.df but df and h.df take values in HΛ m so that dh.f takes values in HΛ m also. Therefore dh takes values in HΛ m i.e. h is a holomorphic integral curve of HΛ m which takes values in H and sat- isfies h(o) = 1.G0. This is in fact equivalent to say that h is a holomorphic integral curve of HΛ m ∗ such that h(o) = 1.G0 (then h ∈ H automatically). Conversely, any holomorphic integral curve of HΛ m ∗ such that h(o) = 1.G0 satisfies that d(h.f) takes values in HΛ m, and thus f := h.f is in Holm (L, ΛGτ /G0) and in the same fibre as f. We have proved that the fibres of πm are all isomorphic to the set of holomorphic integral curves h of HΛ m ∗ such that h(o) = 1.G0. Moreover, proceeding as above we prove that the previous space Holm ∗ (L, ΛGτ /G0) of holomorphic maps is a subgroup of C∞ (L, H) (take for f an element of this space, i.e. in the fibre of the constant map 1.G0, and then applying what precedes proves that this space is stable by multiplication of two elements). This completes the proof. 2.2.4. The decreasing sequence (Syst(m, τp ))p/k . We will call m-th g-system, the m-th (g, Id)-system (i.e. u = (u0, . . . , um) takes values in (gC )m+1 , in defini- tion 2.2.1). Any solution of the m-th (g, τ)-system is solution of the m-th g-system.More precisely, the m-th (g, τ)-system is the restriction to ⊕m j=0gC −j(τ) of the m-th g- system. More generally, for any p ∈ N∗ such that p divides k , the m-th (g, τ)-system is the m-th (g, τp )-system restricted to ⊕m j=0g−j(τ), or equivalently - in terms of αλ ∈ ΛgC τp - restricted to ΛgC τ . 2.3. The minimal determined case. We study here the elliptic system (Syst(m)) in the minimal determined case and by the way its subcase the primitive case. Let us recall again that in this case, we can consider that the system (Syst(m)) deals only with Maurer-cartan forms α and consequently also with geometric maps f. Then we have to translate the equations on α into geometric conditions on f. This is what we will begin to do now.
- 54. m-TH ELLIPTIC INTEGRABLE SYSTEM ASSOCIATED TO k -SYMMETRIC SPACE 39 The minimal determined case splits into two cases. 2.3.1. The even minimal determined case: k = 2k and m = k. Let us recall the following decomposition gC = gC −(k−1) ⊕ . . . ⊕ gC −1 ⊕ gC 0 ⊕ gC 1 ⊕ . . . ⊕ gC k−1 ⊕ gC k . It is useful for the following to keep in mind that k = −k mod 2k. Proposition 2.3.1. The system (Syst(k, τ)) can be written (2.16) ⎧ ⎪ ⎨ ⎪ ⎩ α −j = 0, 1 ≤ j ≤ k − 1 (Hj) dα + 1 2 [α ∧ α] = 0 (MC) ¯ ∂α −k + [α 0 ∧ α −k] = 0 (Sk) . More precisely the equations (Sj), 0 ≤ j ≤ k − 1, of (Syst(k, τ)) are respectively the projection on gC −j, 0 ≤ j ≤ k − 1, of (MC) (owing to the holomorphicity conditions (Hj) given by proposition 2.2.3). Proof. The equation (Sj), for 1 ≤ j ≤ k − 1, is written in terms of α: ¯ ∂α −j + k−j i=0 [α i ∧ α −j−i] = 0, according to definition 2.2.1. Since we have α −j = 0, 1 ≤ j ≤ k − 1, according to proposition 2.2.3, this equation is equivalent to (2.17) dα−j + k−j i=0 [αi ∧ α−j−i] = 0 which is nothing but the projection of (MC) on gC −j. Indeed, we have 1 2 [α∧α]gC −j = 1 2 ⎛ ⎝ i+l=−j [α i ∧ α l]+ i+l=−j [α i ∧ α l ] ⎞ ⎠= i+l=−j [α i ∧α l]= k i=−k+1 [α i ∧α −j−i] = k−j i=0 [α i ∧ α −j−i] where we have used, in the last line, the holomorphicity conditions: α i = 0 if −k + 1 ≤ i ≤ −1 and α −j−i = 0 if k − j ≤ i ≤ k. Now, since α is real (i.e. is g-valued), this equation (2.17) is also equivalent to its complex conjugate i.e. the projection of (MC) on gC j . Moreover, the equation (S0) is written in terms of α: ¯ ∂α 0 + ∂α 0 + k j=0 [α −j ∧ α j ] = 0 and moreover, using the holomorphicity conditions, we have k j=0 [α −j ∧ α j ] = k−1 j=0 [α−j ∧ αj] + 1 2 [α−k ∧ αk] = 1 2 [α ∧ α]g0 which proves that (S0) is equivalent to the projection of (MC) on g0. Finally, the equation (Sk) is written in terms of α as in (2.16). This completes the proof.
- 55. Random documents with unrelated content Scribd suggests to you:
- 56. Λουκιανού (Δίκη φωνηέντων)· αλλά ησύχασε γιατί ο Πίνδαρος μ' όλον τούτο μένει πάντα ο ίδιος για καθέναν· ο ίδιος για με, οπού βρίσκω την τέχνην οπού είναι, ο ίδιος για σε, οπού ξανοίγεις ταις οξείαις οπού δεν λείπουν . . . βλέπω ένα χαμόγελο εις τα χείλα των ξένων αλλά δεν το κάνουν τόσο πικρό, γιατί βέβαια θυμούνται και τα δικά τους. Ως προς την έννοια των στροφών 24 και 25 (του Ύμνου), μέσα στην Κερκυραϊκήν έκδοση (ΣΕ. Κ. 1859) βρίσκεται και η εξής αυτόγραφη σημείωση του ποιητή: «Για να μη με ξανασκοτίσουν οι φλύαροι, οι οποίοι μη γνωρίζοντας τη φύση της Τέχνης δεν ημπορούν να κρίνουν σωστά τα μέσα οπού μεταχειρίζεται, φανερώνω, ότι όχι μόνον η Μεγάλη Βρεταννία δεν είναι χτυπημένη από τούτη τη στροφή, αλλά παρασταίνεται δυνατή, και άγρυπνη εις τα μεγάλα συμβεβηκά του κόσμου. Νά, γυμνός ο στοχασμός, οπού επαράστησα με μίαν εικόνα. Η Μεγάλη Βρεταννία αλαφιάζεται εκείνη τη στιγμή μήπως τα κινήματά μας επροέρχονταν(!) από τη Ρωσσία, και της λέει· θέλεις να πάρης εσύ 'ς τη στεριά, δύνουμαι να πάρω κ' εγώ 'ς το πέλαγο, κ' ετοιμάζομαι. — Η αλήθεια είναι που επήρε λάθος, και χαίρομαι.» ΕΙΣ ΤΟ ΘΑΝΑΤΟ ΤΟΥ ΛΟΡΔ ΜΠΑΙΡΟΝ Ποίημα λυρικό. 1 Λευθεριά, για λίγο πάψε Να χτυπάς με το σπαθί. Τώρα σίμωσε και κλάψε Εις του Μπάιρον το κορμί· 2 Και κατόπι ας ακλουθούνε Όσοι επράξανε λαμπρά· Αποπάνου του ας χτυπούνε Μόνον στήθια ηρωικά.
- 57. 3 Πρώτοι ας έλθουνε οι Σουλιώταις, Και απ' το Λείψανον αυτό Ας μακραίνουνε οι προδόταις, Και απ' τα λόγια οπού θα πω. 4 Φλάμπουρα, όπλα τιμημένα, Ας γυρθούν κατά τη γη, Καθώς ήτανε γυρμένα Εις του Μάρκου τη θανή, 5 Που 'βαστούσε το μαχαίρι, Όταν του λειψε η ζωή, Μέσ' 'ς το ανδρόφονο το χέρι, Και δεν τ' άφινε να βγη. [56] 6 Αναθράφηκε ο γενναίος 'Σ των αρμάτων την κλαγγή· Τούτον έμπνευσε, όντας νέος, Μία θεά μελωδική. — 7 Με ταις θείαις της αδελφάδες Εστεκότουν σιωπηλή, Ενώ αυξαίνανε οι λαμπράδαις 'Σ του Θεού την κεφαλή, 8 Που εμελέτουνε τη Χτίση· Και ότι εβγήκε η προσταγή, Οπού εστένεψε τη Φύση Αιφνιδίως να φωτιστή, 9
- 58. Με τα μάτια ακολουθώντας Το νεογέννητο το φως, Και 'σε δαύτο αναφτερώντας, Της εξέβγαινε ο ψαλμός 10 Απ' τ' αθάνατο το στόμα, Και απομάκραινε η βροντή, Που το Χάος έκανε ακόμα 'Σ την ογλήγορη φυγή, 11 Έως που ολόκληρον εχάθη 'Στου Ερέβου τη φυλακή, Οπού απλώθηκε και εστάθη Σα 'ς την πρώτη του πηγή. 12 — Ψάλλε, Μπάιρον, του λαλούσε, Όσαις βλέπεις ομορφιαίς· Και κειος, που εκρυφαγροικούσε Ανταπόκριση μ' αυταίς, 13 Βάνεται, ταις τραγουδάει Μ' ένα χείλο αρμονικό, Και τα πάθη έτσι σου 'γγιάει, Που τραγούδι πλέον ψηλό 14 Δεν ακούστηκεν, απ' ότα (2) Έψαλ' ο Άγγλος ο τυφλός Τ' αγκαλιάσματα τα πρώτα Που έδωσ' άντρας γυναικός. 15
- 59. Συχνά εβράχνιασε η μιλιά του Τραγουδώντας λυπηρά, Πώς 'ς τον ήλιον αποκάτου Είναι λίγη ελευθεριά. 16 «Κάθε γη» παραπονειέται, (3) »Έσκλαβώθηκε, — είνε μια, »Οπού ο άνθρωπος τιμειέται, »Από δώθενε μακριά, 17 »Την οποία χτυπάει το νάμα »Σύνορα τ' Ατλαντικό· »Μετανόνει εν τω άμα »Όποιος πάει με στοχασμό 18 »Τη γλυκειάν Ελευθερία »Να της βλάψη από κοντά· »Το δοκίμασεν η Αγγλία! »Κάνεις πλέον ας μην κοτά.» 19 Και ότι βούλεται να φύγη (4) Εκεί πέρα ο Ποιητής, Ανεπόλπιστα ξανοίγει Εσέ εδώ να πεταχτής. 20 Επετάχτηκες! Μονάχη· Χωρίς άλλος να σου πη· Τώρ α αρ χί νησ ε τ η μ άχη, Και γω πλάκωσα μαζί. 21 Να σ' το πη, και να σε ρήξη 'Στων Τουρκών ταις τουφεκιαίς
- 60. Ασυντρόφιαστη, αν ξανοίξη Ταις περίστασαις δειναίς, 22 Κι' αν ταις εύρη ευτυχισμέναις, Νάλθη αντίς για τον εχθρό, Μ' άλλαις άλυσαις φτειασμέναις Αποκάτου απ' το Σταυρό, 23 [57] Πούχε λάβη 'ς ταις αγκάλαις Από μας, κ' είχε θεούς Αστραπαίς, ανεμοζάλαις, Και βρονταίς και ποταμούς. 24 Μόνον τ' αδικοσφαμένα Τα παιδιά σου, στρυμωχτά, Με τα χέρια τσακισμένα Σε εσπρώξανε ομπροστά, 25 Και Συ εχύθηκες, πετώντας Μία ματιά 'ς τον Ουρανό, Που τα δίκια σου θωρώντας, Απεκρίθηκε· Είμαι δω. 26 [58] Και χτυπώντας ξεθυμαίνει Εις το πέλαγο, εις τη γη, Η ρομφαία σου πυρωμένη Οχ την Άπλαστη Φωνή, 27 Και θαυμάσια τόσα πράχτει, Οπού οι Τύραννοι της γης
- 61. 'Σ εσέ κίνησαν με άχτι, Όμως έστρεψαν ευθύς. 28. Χαίρε! Κι' όποιος σε μισάει, Και πικρά σε λοιδορεί, Ευτυχία να πιθυμάη, Και ποτέ να μη την δη· 29 Και να κλαίη πως ήλθε η ώρα Η πατρίς του να δεθή Με τα σίδερα, που τώρα Πας συντρίβοντας Εσύ. 30 Χαίρου ωστόσο όλους τους τόπους, Που εξανάλαβαν γοργά Πάλι ελεύθερους ανθρώπους· Και του Μπάιρον τη χαρά 31 Χαίρου, ανάμεσα 'ς τα άλλα Πράγματα που σε τιμούν, Οι Μεγάλοι τα μεγάλα, Που τους μοιάζουνε, αγαπούν. 32 Βλέποντάς σε αναγαλλιάζει Η θλιμμένη του ψυχή, Και του λέγει· Ό π λα φωνάζε ι Τώρ α η Ε λλάδα· πάμε ε κε ί . 33 Και κινάει να σ' απαντήση· Και η Φήμη του Ποιητού,
- 62. Που τον κόσμο είχε γυρίση, Και τη δέχτηκαν παντού, 34 Μπροστοπάταε, να σε κράξη Με όνομα τόσο γλυκύ, Που όποιο μάτι σε κυττάξη Σε ξανοίγει πλέον σεμνή. 35 Τον ακλούθησεν ο πλούτος, Θείος 'ς τα χέρια του καλού, Και κακόπραχτος, αν ούτως Και είν' 'ς τα χέρια του κακού. 36 Μ' ένα βλέμμα οπού φονεύει Τα φρονήματα τα αισχρά, Τρομερή τον συντροφεύει, Στέκοντάς του εις τη δεξιά, 37 Και όντας άφαντη 'ς τους άλλους, Του Αλκαίου η σκιά, Και τους ώμους τους μεγάλους Λίγο γέρνοντας, κρυφά 38 Λόγια αθάνατα του λέει, Με τα οποία 'ς τα σωθικά Το θυμό του ξανακαίει Εναντίον 'ς την αδικιά· 39 Θυμόν, τρόμο όλον γεμάτον, Που νικάει την ταραχή
- 63. Των βροντοκραυγών αρμάτων, Και πετειέται ολού με ορμή, 40 Και του τύραννου χτυπάει Τη βουλή, και την ξυπνά, 'Σ τη στιγμή που μελετάει Των λαών τη συφορά. 41 Μόνον άκουε του Κοράκου (5) Της Αυστρίας το κραυγητό, Που δεν έκρωξε του κάκου, Και επεθύμαε το κακό. — 42 Ομοίως έστρεφεν η Μοίρα, (6) Που είχε πάντοτε· σταθή Μέσ' 'ς της Κόλασης τη θύρα Με το Κρίμα ανταμωτή, 43 Έστρεφε κατά τη Χτίση, Γιατί εμύριζε νεκρή Μυρωδία, που χε σκορπίση Η πικρή μεταβολή· 44 Και από τ' άπειρο διάστημα Αντισήκωνε ψηλά Το μιαρό της το ανάστημα, Να χαρή τη μυρωδιά. 45 — 'Σ την Ελλάδα χαροκόπι Γιατί Εκείνον, που ζητεί,
- 64. Βλέπει νάρχεται, και οι τόποι Που η σκλαβιά καταπατεί, 46 Χαμηλή την κεφαλή τους, Αγροικώντας τη βουή, Εδακρύζαν, και οι δεσμοί τους Τους εφάνηκαν διπλοί. 47 Αλλά αμέσως όλοι οι άλλοι Που είχαν ελευθερωθή, Και έχουν δάφνη 'ς το κεφάλι Που δε θέλει μαραθή, 48. Ταις σημαίαις τους ξεδιπλώνουν, Και ταις δάφναις που φορούν Χαιρετώντας τον σηκώνουν, Και μ' αυταίς τον προσκαλούν. 49 Πού θα πάη; Βουνά και λόγγοι Και λαγκάδια αϊλογούν. Που θα πάη; — 'Σ το Μεσολόγγι, Και άλλοι ας μη ζηλοφθονούν. 50 Τέτοιο χώμα, απ' την ημέρα (7) Τη μεγάλη του Χριστού, Που είχε φέρη απ' τον αιθέρα Τιμή εμάς και δόξα Αυτού, 51 Είν' ιερό προσκυνητάρι, Και δε θέλει πατηθή
- 65. Από βάρβαρο ποδάρι, Πάρεξ όταν χαλαστή. 52 Δεν ήταν τη μέρα τούτη Μοσχολίβανα, ψαλμοί. Νά, μολίβια, νά, μπαρούτι, Νά, σπαθιών λαμποκοπή. 53 'Σ τον αέρα ανακατώνονται Οι σπιθόβολοι καπνοί, Και από πάνου φανερώνονται Ήσκιοι θείοι πολεμικοί· 54 Και είναι αυτοί, που πολεμώντας Εσκεπάσανε τη γη, Πάνου εις τ' άρματα βροντώντας Με το ελεύθερο κορμί 55 [59] Και αγκαλιάσματα εκεί πλήθια, Δάφναις έλαβαν, φιλιά, Όσα ελάβανε εις τα στήθια Βόλια τούρκικα σπαθιά. 56 Όλοι εκείνοι οι πολεμάρχοι Περιζώνουνε πυκνοί Την ψυχή του Πατριάρχη, Που τον πόλεμο ευλογεί· 57 Και αναδεύονται, και γέρνουν, Και εις το πρόσωπο ιλαροί,
- 66. Χεραπλώνουνε και παίρνουν Από τη σπιθοβολή. 58 Εδώ βλέπει αντρειωμένα Να φρονούν παρά ποτέ· Και όλος έρωτα για σένα Προσηλώνεται 'ς εσέ· 59 Το πουλί, που βασιλεύει Πάνου εις τ' άλλα τα πουλιά, Γληγορώτατα αναδεύει Τα αιθερόλαμνα φτερά, 60 Τρέχει, χάνεται, και πίνει Τόλμην πίνει ο οφθαλμός φως τ' άστρον, οπού χύνει Κύματα άφθαρτα φωτός. 61 Πλανημένη η φαντασιά του Μέσ' 'ς το μέλλον το αργό, Που προσμένει τ' όνομά του Να το κάμη πλέον λαμπρό, 62 Ολοφλόγιστη πηδάει Εισέ μία ματιού ροπή Στρέφει απέκει και κυττάει· — Ανεκδιήγητη αντηχεί 63 Απ' του κόσμου όλου τα πέρατα Του Καιρού η χλαλοή,
- 67. Και διηγώντας του τα τέρατα Του χτυπάει την ακοή· 64 Έθνη που άλλα φοβερίζουν, Φωναίς, θρόνοι δυνατοί· Άλλοι πέφτουνε, άλλοι τρίζουν, Και άλλοι ατάραχτοι και ορθοί· 65 Από φόβο και από τρόμο, Από βάρβαρους δεσμούς, Που ναι σκόρπιοι εις κάθε δρόμο, Και από μύριους υβρισμούς, 66 Βγαίνει, ανάμεσα 'ς τους κρότους Των γενναίων που την παινούν, Και κυττούνται ανάμεσό τους Για το θαύμα που θωρούν, 67 Μία Γυναίκα, που χε βάλη Μέσ' 'ς τα βάσανα ο καιρός, Ξαναδείχνοντας τα κάλλη Που της έσβυσε ο ζυγός, 68 Μόνον έχοντας για σκέπη Τα τουφέκια τα εθνικά, Και το χαίρεται να βλέπη Πώς και Αυτός την ακλουθά. 69 Αχ! συνέρχεται . . . ξανοίγει Ερινύαν φαρμακερή,
- 68. Οπού αγιάτρευτην ανοίγει Της Ελλάδας μίαν πληγή· 70 Ερινύαν από τα χθόνια Που η Ελπίδα απαρατά· Η θεομίσητη Διχόνοια Που τον άνθρωπο χαλνά. 71 Αφού εδιώχτηκε από τ' άστρα Οπού ετόλμησε να πα, Πάει 'ς τους κάμπους, πάει 'ς τα κάστρα, Χωρίς ναύρη δυσκολιά. 72 Και κρατώντας κάτι φίδια Που είχε βγάλη απ' την καρδιά, Και χτυπώντας τα πιτήδεια Εις τους Ελληνας, περνά· 73 Και όχι πλέον τραγούδια νίκης Ωσάν πρώτα, ενώ τυφλά, Με το τρέξιμο της φρίκης, Τουρκικά άλογα πολλά 74 Ετσακίζανε τα χνάρια 'Σ την απέλπιστη φυγή, Και εγκρεμίζαν παλληκάρια Του γκρεμνού από την κορφή 75 Όχι πλέον, όχι τα δυνα- τά στοιχεία να μας θωρούν,
- 69. Και να οργίζωνται και εκείνα Και για μας να πολεμούν· (8) 76 Αλλά πάει 'ς τους νόας μία θέρμη, Που είναι αλλιώτικη απ' αυτή, Οπού εσκόρπησε 'ς την έρμη Χιο του Τούρκου η πιβουλή, (9) 77 Όταν τόσοι επέφταν χάμου, Και με λόγια απελπισιάς, Κόψε με, έλεγαν, Αγά μου, Και τους έκοβεν ο Αγάς. 78 Όμως θέρμη ποίος υβρίζει Τον καλύτερο, και ποιος Λόγια ανόητα ψιθυρίζει· Άλλος στέκεται οκνηρός· 79 Άλλος παίρνει το ποτήρι Αποκάτου απ' την ελιά, Ωσάν νά τουν πανηγύρι, Με τα πόδια διπλωτά· 80 Και άλλοι, αλίτηροι! χτυπώντας Πέφτουνε 'ς τον αδελφό, Και παινεύονται, θαρρώντας Πως εχτύπησαν εχθρό. 81 Και τους φώναξε· «Φευγάτε Τ'ς Ερινύας την τρικυμιά·
- 70. Ω! τι κάνετε; Πού πάτε; Για φερθήτε ειρηνικά· 82 »Γιατί αλλιώς θε να βρεθήτε »Ή με ξένο βασιλιά, »Ή θα καταφανισθήτε »Από χέρια αγαρηνά.»(10) 83 Αφού εδώ 'ς την παλαιά σου Κατοικία και άλλη φορά Με διχόνοιαις τα παιδιά σου Σου ετοιμάσανε εξοριά, 84 Από τότες οπού εσώθη 'Σ την Ελλάδα ο Στρατηγός, Οπού ο Έλληνας ειπώθη (Και τώρα όχι) ο στερινός, 85 Έως που ο κόσμος εβαστούσε Τον απάνθρωπον Αλή, Που όσον αίμα και αν ρουφούσε Τόσο εγύρευε να πιή, 86 [60] Επερνούσαν οι αιώνες Ή σε ξένη υποταγή, Ή με ψεύτικαις κορώναις, Ή με σίδερα και οργή· 87 Και ήλθε τότες και επερπάτει (11) Όπου επάταγες Εσύ,
- 71. Και του δάκρυζε το μάτι, Και επιθύμαε να Σε ιδή, 88 Κ' έλεε· Π ό τ ε έ ρ χε σ αι π άλι Και δεν είναι αληθινό, Πως μας είχε άδικο βάλη Με βρισιαίς και με θυμό. (12) 89 Εζωγράφιζαν οι στίχοι Τον γαλάζιον ουρανό, Και εκλαιόνταν με την τύχη, Και με τ' άστρο το κακό, 90 Εις το οποίον έχει να σκύψη Κάθε δύναμη θνητή, Και η πατρίδα του να στρίψη Παντελώς δεν ημπορεί. 91 Τώρα αθάμπωτη έχει δόξα, Και με φέρσιμο τερπνόν Βλέπει αδύνατα τα τόξα Των αντίζηλων εθνών 92 Και λαούς αλυσοδένει, Και εις τα πόδια τους πατεί, Και το πέλαγο σωπαίνει Αν του σύρη μία φωνή· 93 Τέχναις, άρματα, σοφία, Τήνε κάνουν δοξαστή,
- 72. Όμως θα βρούνε ευκαιρία Να τη φθείρουνε οι καιροί, 94 Και να ιδή το ριζικό της Καθώς είναι η καταχνιά, Που εις το κλίμα το δικό της Κρύβει την αστροφεγγιά. 95 «Πού είν'» θα λένε σαστισμένοι »Το Λεοντάρι το Αγγλικό; »Είναι η χήτη του πεσμένη, »Και το μούγκρισμα βουβό.» 96 Αλλ' η Ελλάς να ξαναζήση Ήταν άξια, και να ιδή Ο ερχομός να την τιμήση Του υψηλότατου Ποιητή. 97 Έστεκε 'ς το μισημένο Το ζυγό μ' αραθυμιά· Το ποδάρι είχε δεμένο, Αλλά ελεύθερη καρδιά. 98 Εκαθότουνε εις τα όρη Ο Σουλιώτης ξακουστός· Να τον διώξη δεν ημπόρει Πείνα, δίψα, και αριθμός. (13) 99 Συχνά σπώντας τα θηκάρια Με τα χέρια τα λιγνά,
- 73. Ορμούν 'ς άπειρα κοντάρια Ταις γυναίκες των συχνά 100 Μεγαλόψυχα τραβάει Το ίδιον αίσθημα τιμής, Που κυττώντας τον Κομβάυ Είχε ο ανδρείος Τραγουδιστής. (14) 101 Ταις εμάζωξε εις το μέρος Του Τσαλόγγου το ακρινό Της ελευθεριάς ο έρως, Και ταις έμπνευσε χορό· (15) 102 Τέτοιο πήδημα δεν το είδαν Ούτε γάμοι, ούτε χαραίς, Και άλλαις μέσα τους επήδαν Αθωότεραις ζωαίς· 103 Τα φορέματα εσφυρίζαν Και τα ξέπλεκα μαλλιά, Κάθε γύρο που εγυρίζαν Από πάνου έλειπε μια· 104 Χθες γόγγυσμα κι' αντάρα Παρά εκείνη μοναχά, Οπού έκαναν με την κάρα, Με τα στήθια, 'ς τα γκρεμά. 105 'Σ τα ίδια όρη εγεννηθήκαν Και τα αδάμαστα παιδιά,
- 74. Που την σήμερο εχυθήκαν Πάντα οι πρώτοι 'ς τη φωτιά. 106 Γιατί, αλίμονον! γυρίζοντας Τ'ς ηύρε ο Μπάιρον σκυθρωπούς; Εγυρεύανε δακρύζοντας Τον πλέον ένδοξο απ' αυτούς. (16) 107 Όταν 'ς της νυχτός τα βάθη Τα πάντα όλα σιωπούν. Και εις τον άνθρωπο τα πάθη, Πούναι ανίκητα, αγρυπνούν, 108 Και γυρμένοι εις το πλευρό τους Οι στρατιώταις του Χριστού, Μύρια βλέπουν 'ς τ' όνειρό τους Ξεψυχίσματα του εχθρού· 109 Αυτός άγρυπνος στενάζει, Και εις την πλάκα την πικρή, Που τον Μπότσαρη σκεπάζει, Για πολλή ώρα αργοπορεί· 110 Έχει πλάγιασμα θανάτου Και άλλος άντρας φοβερός (17) Εις τα πόδια του αποκάτου, Και είναι αντίκρυ του ο ναός. 111 Ακριβό σαν την ελπίδα Που έχει πάντοτε ο θνητός.
- 75. Γλυκοφέγγει απ' τη θυρίδα Τ'ς Άγιας Τράπεζας το φως· 112 Μέσαθε έπαιρνε ο αέρας Με δροσόβολη πνοή Το λιβάνι της ημέρας, Και του τό φερνε ως εκεί. 113 Δεν ακούς γύρου πατήματα· Mόν' τον ήσκιο του θωρείς, Οπού απλώνεται 'ς τα μνήματα, Έρμος, άσειστος, μακρύς, 114 Καθώς βλέπεις και μαυρίζει Ήσκιος νέου κυπαρισσιού, Αν την άκρη του δε 'γγίζει Αύρα ζέφυρου λεπτού. 115 Πες μου, Ανδρείε, τι μελετούνε Οι γενναίοι σου στοχασμοί, Που πολληώρα αργοπορούνε Εις του Μάρκου την ταφή; 116 Σκιάζεσαι ίσως μη χουμήσουν Ξάφνου οι Τούρκοι το πρωί, (18) Και το στράτευμα νικήσουν, Που έχει ανίκητην ορμή; 117 Σκιάζεσαι τους Βασιλιάδες, Που έχουν Ένωσιν Ιερή,
- 76. Μη φερθούνε ωσάν Πασάδες 'Σ τον Μαχμούτ εμπιστευτοί; 118 Ή σου λέει 'ς τα σπλάχνα η φύσις Μ' ένα κίνημα κρυφό, — »Την Ελλάδα θε ν' αφήσης, »Για να πας 'ς τον Ουρανό» — ; 119 Βγαίνει μάγεμα απ' τη στάχτη Των Ηρώων, και τον βαστά, Και τη θέληση του αδράχτει· Τότε αισθάνεται με μια 120 Την αράθυμη ψυχή του, Που με φλόγα αναζητεί Να του σύρη το κορμί του Σε φωτιά πολεμική. — 121 Του πολέμου ένδοξοι οι κάμποι! Είδ' η Ελλάδα τολμηρά Και το Σοφοκλή να λάμπη Μέσα 'ς την αρματωσιά· 122 Και είδε Αυτόν, που παρασταίνει Μαζωμένους τους Εφτά 'Σ την ασπίδα αιματωμένη, Οπού ωρκόνονταν φριχτά· 123 Ετραγούδααν προθυμότερα Ταις ωδαίς του τα παιδιά,
- 77. Και αισθανόντανε, αντρειότερα 'Στην ανήλικη καρδιά· 124 Και τα μάτια τους γελούσαν, Μάτια μαύρα ως την ελιά Των μερτιών, οπού βαστούσαν Τραγουδώντας ταις γλυκά. 125 — 'Σ τη φωτιά! και θρέφει ελπίδα Να νικήση να ημπορή Να επιστρέψη 'ς την Πατρίδα, Το κοράσιο του να ευρή· 126 Να του λέγη μ' ένα δάκρυ· «Χαίρου, τέκνο μου ακριβό, »Εις του στήθους μου την άκρη «Ελαβώθηκα και εγώ. 127 «Βάλε, φως μου, την παλάμη »Εις τα στήθια του πατρός· »Νά, την ζώνη που είχε κάμη »Κόρη τούρκισσα του αντρός.» 128 Και το πέλαγο αγναντεύει Ίσως τώρα η κορασιά, Και ξεφάντωση γυρεύει Με τραγούδια τρυφερά. 129 «Τον γονιό μου, Πρόνοια Θεία, »Κάμε τόνε νικητή,
- 78. »Εις τα χώματα, 'ς τα οποία »Η γυναίκα απαρατεί 130 »Τα στολίδια, τον καθρέφτη, »Και αποκάτου απ' το βυζί »Ζώνεται άρματα, και πέφτει »Όπου κίνδυνο θωρεί· 131 »Κάμε Εσύ με την μητέρα »Τη γλυκειά μου να ενωθή· (19) »Έλα γλήγορα, πατέρα, »Όλη η Αγγλία σε καρτερεί. 132 »Το καράβι πότε αράχνει »Εισέ θάλασσα αγγλική; »Μου σπαράζουνε τα σπλάχνη, »Οπού μου έκαμες εσύ. 133 »Πες, πότ' έρχεσαι;» . . . Ολοένα Είν' το πλοίο του 'ς τα νερά, Που φλοισβίζουνε σχισμένα, Και ποσώς δε τ' αγροικά. 134 Ποίος αλίμονον! μας δίνει Μίαν αρχή παρηγοριάς; Απ' αυτόν δε θε να μείνη Μήτε η στάχτη του με μας· 135 Θα την έχουν άλλοι! . . . Ω! σύρε, Σύρε, Μπάιρον, 'ς το καλό·
- 79. Ύπνος έξαφνα σ' επήρε, Που δεν έχει ξυπνημό· 136 Είναι αδιάφορο, δε βλάβει, Αν εκεί σιμοτινό Πλέξη ή τούρκικο καράβι, Ή καράβι ελληνικό. 137 Άκου, Μπάιρον, πόσον θρήνον Κάνει, ενώ σε χαιρετά, Η Πατρίδα των Ελλήνων Κλαίγε, κλαίγε, Ελευθεριά. 138 Γιατί εκείτεταν 'ς την κλίνη, Και του εβάραινε πολύ Πως για πάντα είχε να μείνη, Και από Σε να χωριστή· 139 [61] Αρχινάει του ξεσκεπάζει Άλλον κόσμο ο λογισμός, Και κάθε άλλο σκοταδιάζει, Και του κρύβεται απ' εμπρός. 140 Αλλά αντίκρυ από τα πλάσματα Του νοός τα αληθινά, Του προβαίνουν δυο φαντάσματα ολοζώντανα και ορθά· 141 Η ακριβή του θυγατέρα, Καθώς έμεινε μικρή, (20)
- 80. Welcome to our website – the perfect destination for book lovers and knowledge seekers. We believe that every book holds a new world, offering opportunities for learning, discovery, and personal growth. That’s why we are dedicated to bringing you a diverse collection of books, ranging from classic literature and specialized publications to self-development guides and children's books. More than just a book-buying platform, we strive to be a bridge connecting you with timeless cultural and intellectual values. With an elegant, user-friendly interface and a smart search system, you can quickly find the books that best suit your interests. Additionally, our special promotions and home delivery services help you save time and fully enjoy the joy of reading. Join us on a journey of knowledge exploration, passion nurturing, and personal growth every day! ebookbell.com