![Preface: why projective ?
Metrical geometry is a part of descriptive geometry1, and de-
scriptive geometry is all geometry.
Arthur Cayley
On October 5-th 2001, the authors of this book typed in the word
“Schwarzian” in the MathSciNet database and the system returned 666 hits.
Every working mathematician has encountered the Schwarzian derivative at
some point of his education and, most likely, tried to forget this rather scary
expression right away. One of the goals of this book is to convince the reader
that the Schwarzian derivative is neither complicated nor exotic, in fact, this
is a beautiful and natural geometrical object.
The Schwarzian derivative was discovered by Lagrange: “According to
a communication for which I am indebted to Herr Schwarz, this expression
occurs in Lagrange’s researches on conformable representation ‘Sur la con-
struction des cartes géographiques’ ” [117]; the Schwarzian also appeared in
a paper by Kummer in 1836, and it was named after Schwarz by Cayley. The
main two sources of current publications involving this notion are classical
complex analysis and one-dimensional dynamics. In modern mathematical
physics, the Schwarzian derivative is mostly associated with conformal field
theory. It also remains a source of inspiration for geometers.
The Schwarzian derivative is the simplest projective differential invari-
ant, namely, an invariant of a real projective line diffeomorphism under the
natural SL(2, R)-action on RP1
. The unavoidable complexity of the for-
mula for the Schwarzian is due to the fact that SL(2, R) is so large a group
(three-dimensional symmetry group of a one-dimensional space).
Projective geometry is simpler than affine or Euclidean ones: in pro-
jective geometry, there are no parallel lines or right angles, and all non-
degenerate conics are equivalent. This shortage of projective invariants is
1
By descriptive geometry Cayley means projective geometry, this term was in use in
mid-XIX-th century.
vii](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-11-320.jpg)
![1.2. DISCRETE INVARIANTS AND CONFIGURATIONS 5
the equation
t4
+ a1t2
+ a2t + a3 = 0. (1.1.3)
This 1-parameter family of planes envelops a surface called the swallow tail
and shown in figure 1.4. This developable surface consists of the tangent
lines to the curve γ∗. Note the cusp of γ∗ at the origin.
Figure 1.4: Swallow tail
Comment
The study of polynomials (1.1.3) and figure 1.4 go back to the XIX-th cen-
tury [118]; the name “swallow tail” was invented by R. Thom in mid XX-th
century in the framework of the emerging singularity theory (see [16]). The
swallow tail is the set of polynomials (1.1.3) with multiple roots, and the
curve γ∗ corresponds to polynomials with triple roots. This surface is a
typical example of a developable surface, i.e., surface of zero Gauss curva-
ture. The classification of developable surfaces is due to L. Euler (cf.[193]):
generically, such a surface consists of the tangent lines of a curve, called the
edge of regression. The edge of regression itself has a singularity as in figure
1.4.
Unlike the Plücker formula of classic algebraic geometry, the result of
Exercise 1.1.2 is surprisingly recent; it was obtained by Fabricius-Bjerre in
1962 [61]. This result has numerous generalizations, see, e.g., [199, 66].
1.2 Discrete invariants and configurations
The oldest invariants in projective geometry are projective invariants of
configurations of point and lines. Our exposition is just a brief excursion to
the subject, for a thorough treatment see, e.g., [22].](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-19-320.jpg)
![6 CHAPTER 1. INTRODUCTION
Cross-ratio
Consider the projective line RP1
. Every triple of points can be taken to
any other triple by a projective transformation. This is not the case for
quadruples of points: four points in RP1
have a numeric invariant called the
cross-ratio. Choosing an affine parameter t to identify RP1
with R ∪ {∞},
the action of PGL(2, R) is given by fractional-linear transformations:
t 7→
at + b
ct + d
. (1.2.1)
The four points are represented by numbers t1, t2, t3, t4, and the cross-ratio
is defined as
[t1, t2, t3, t4] =
(t1 − t3)(t2 − t4)
(t1 − t2)(t3 − t4)
. (1.2.2)
A quadruple of points is called harmonic if its cross-ratio is equal to −1.
Exercise 1.2.1. a) Check that the cross-ratio does not change under trans-
formations (1.2.1).
b) Investigate how the cross-ratio changes under permutations of the four
points.
A B C D
A B C D
a b c
d
' ' ' '
Figure 1.5: Cross-ratio of lines: [A, B, C, D] = [A0, B0, C0, D0] := [a, b, c, d]
One defines also the cross-ratio of four concurrent lines in RP2
, that is,
four lines through one point. The pencil of lines through a point identifies
with RP1
, four lines define a quadruple of points in RP1
, and we take their
cross-ratio. Equivalently, intersect the four lines with an auxiliary line and
take the cross-ratio of the intersection points therein, see figure 1.5.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-20-320.jpg)
![8 CHAPTER 1. INTRODUCTION
Figure 1.7: Pascal and Brianchon theorems
A D
C
B
P P1
Figure 1.8: Steiner theorem
asserts that this cross-ratio is independent of the choice of point P:
[(PA), (PB), (PC), (PD)] = [(P1A), (P1B), (P1C), (P1D)]
in figure 1.8.
Comment
In 1636 Girard Desargues published a pamphlet “A sample of one of the
general methods of using perspective” that laid the foundation of projective
geometry; the Desargues theorem appeared therein. The Pappus configura-
tion is considerably older; it was known as early as the III-rd century A.D.
The triple of lines in figure 1.6 is a particular case of a cubic curve, the
Pappus configuration holds true for 9 points on an arbitrary cubic curve –
see figure 1.9. This more general formulation contains the Pascal theorem
as well. Particular cases of Steiner’s theorem were already known to Apol-](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-22-320.jpg)
![1.3. INTRODUCING SCHWARZIAN DERIVATIVE 9
Figure 1.9: Generalized Pappus theorem
lonius 1. Surprisingly, even today, there appear new generalizations of the
Pappus and the Desargues theorems, see [182, 183].
1.3 Introducing Schwarzian derivative
Projective differential geometry studies projective invariants of functions,
diffeomorphisms, submanifolds, etc. One way to construct such invariants
is to investigate how discrete invariants vary in continuous families.
Schwarzian derivative and cross-ratio
The best known and most popular projective differential invariant is the
Schwarzian derivative. Consider a diffeomorphism f : RP1
→ RP1
. The
Schwarzian derivative measures how f changes the cross-ratio of infinitesi-
mally close points.
Let x be a point in RP1
and v be a tangent vector to RP1
at x. Extend
v to a vector field in a vicinity of x and denote by φt the corresponding local
one-parameter group of diffeomorphisms. Consider 4 points:
x, x1 = φε(x), x2 = φ2ε(x), x3 = φ3ε(x)
1
We are indebted to B. A. Rosenfeld for enlightening discussions on Ancient Greek
mathematics](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-23-320.jpg)
![10 CHAPTER 1. INTRODUCTION
(ε is small) and compare their cross-ratio with that of their images under f.
It turns out that the cross-ratio does not change in the first order in ε:
[f(x), f(x1), f(x2), f(x3)] = [x, x1, x2, x3] − 2ε2
S(f)(x) + O(ε3
). (1.3.1)
The ε2–coefficient depends on the diffeomorphism f, the point x and the
tangent vector v, but not on its extension to a vector field.
The term S(f) is called the Schwarzian derivative of a diffeomorphism
f. It is homogeneous of degree 2 in v and therefore S(f) is a quadratic
differential on RP1
, that is, a quadratic form on TRP1
.
Choose an affine coordinate x ∈ R ∪ {∞} = RP1
. Then the projective
transformations are identified with fractional-linear functions and quadratic
differentials are written as φ = a(x) (dx)2. The change of variables is then
described by the formula
φ ◦ f = f0
2
a(f(x)) (dx)2
. (1.3.2)
The Schwarzian derivative is given by the formula
S(f) =
f000
f0
−
3
2
f00
f0
2
!
(dx)2
. (1.3.3)
Exercise 1.3.1. a) Check that (1.3.1) contains no term, linear in ε.
b) Prove that S(f) does not depend on the extension of v to a vector field.
c) Verify formula (1.3.3).
The Schwarzian derivative enjoys remarkable properties.
• By the very construction, S(g) = 0 if g is a projective transformation,
and S(g ◦ f) = S(f) if g is a projective transformation. Conversely, if
S(g) = 0 then g is a projective transformation.
• For arbitrary diffeomorphisms f and g,
S(g ◦ f) = S(g) ◦ f + S(f) (1.3.4)
where S(g) ◦ f is defined as in (1.3.2). Homological meaning of this
equation will be explained in Section 1.5.
Exercise 1.3.2. Prove formula (1.3.4).](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-24-320.jpg)
![1.3. INTRODUCING SCHWARZIAN DERIVATIVE 13
The monodromy condition (1.3.6) implies that, for the corresponding
Sturm-Liouville operator, one has u(t + 1) = u(t), while the solutions have
the monodromy f
M ∈ SL(2, R), which is a lift of M. To summarize, the space
of projective structures on S1 is identified with the space of Sturm-Liouville
operators with 1-periodic potentials u(t).
Diff(S1
)- and Vect(S1
)-action on projective structures
The group of diffeomorphisms Diff(S1) naturally acts on projective atlases
and, therefore, on the space of projective structures. In terms of the Sturm-
Liouville operators, this action is given by the transformation rule for the
potential
Tf−1 : u 7→ f0
2
u(f) +
1
2
S(f), (1.3.7)
where f ∈ Diff(S1). This follows from Exercise 1.3.4 and formula (1.3.4).
The Lie algebra corresponding to Diff(S1) is the algebra of vector fields
Vect(S1). The vector fields are written as X = h(t)d/dt and their commu-
tator as
[X1, X2] = h1h0
2 − h0
1h2
d
dt
.
Whenever one has a differentiable action of Diff(S1), one also has an action
of Vect(S1) on the same space.
Exercise 1.3.5. Check that the action of a vector field X = h(t)d/dt on
the potential of a Sturm-Liouville operator is given by
tX : u 7→ hu0
+ 2h0
u +
1
2
h000
. (1.3.8)
It is interesting to describe the kernel of this action.
Exercise 1.3.6. a) Let φ1 and φ2 be two solutions of the Sturm-Liouville
equation φ00(t)+u(t)φ(t) = 0. Check that, for the vector field X = φ1φ2 d/dt,
one has tX = 0.
b) The kernel of the action t is a Lie algebra isomorphic to sl(2, R); this
is precisely the Lie algebra of symmetries of the projective structure corre-
sponding to the Sturm-Liouville operator.
Hint. The space of solutions of the equation tX = 0 is three-dimensional,
hence the products of two solutions of the Sturm-Liouville equation span
this space.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-27-320.jpg)
![14 CHAPTER 1. INTRODUCTION
Γ
y1
y2
y1
y2
Figure 1.11: Zeroes of solutions
Sturm theorem on zeroes
The classic Sturm theorem states that between two zeroes of a solution of a
Sturm-Liouville equation any other solution has a zero as well. The simplest
proof is an application of the above identification between Sturm-Liouville
equations and projective structures on S1. Consider the corresponding de-
veloping map γ : S1 → RP1
and its lift Γ to R2. Every solution φ of the
Sturm-Liouville equation is a pull-back of a linear function y on R2. Ze-
roes of φ are the intersection points of Γ with the line y = 0. Since γ is
non-degenerate, the intermediate value theorem implies that between two
intersections of Γ with any line there is an intersection with any other line,
see figure 1.11 and [163] for an elementary exposition.
Comment
The Schwarzian derivative is historically the first and most fundamental
projective differential invariant. The natural identification of the space of
projective structures with the space of Sturm-Liouville operators is an im-
portant conceptual result of one-dimensional projective differential geome-
try, see [222] for a survey. Exercise 1.3.6 is Kirillov’s observation [115].
1.4 Further example of differential invariants: pro-
jective curvature
The second oldest differential invariant of projective geometry is the pro-
jective curvature of a plane curve. The term “curvature” is somewhat mis-
leading: the projective curvature is, by no means, a function on the curve.
We will define the projective curvature as a projective structure on the curve.
In a nutshell, the curve is approximated by its osculating conic which, by](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-28-320.jpg)
![1.4. FURTHER EXAMPLE OF DIFFERENTIAL INVARIANTS: PROJECTIVE CURVATURE17
Exercise 1.4.5. a) Let A be the differential operator corresponding to a
non-degenerate parameterized curve γ(t) in RP2
. Prove that the operator
corresponding to the dual curve γ∗(t) is −A∗.
b) Consider a non-degenerate parameterized curve γ(t) in RP2
and let γ∗(t)
be projectively equivalent to γ(t), i.e., there exists a projective isomorphism
ϕ : RP2
→ RP2∗
such that γ∗(t) = ϕ(γ(t)). Prove that γ(t) is a conic.
l
l
l
l
0
1
2
3
Figure 1.12: Projective curvature as cross-ratio
Projective curvature and cross-ratio
Consider four points
γ(t), γ(t + ε), γ(t + 2ε), γ(t + 3ε)
of a non-degenerate curve in RP2
. These points determine four lines `0, `1, `2
and `3 as in figure 1.12.
Let us expand the cross-ratio of these lines in powers of ε.
Exercise 1.4.6. One has
[`0, `1, `2, `3] = 4 − 2ε2
q(t) + O(ε3
). (1.4.8)
This formula relates the projective curvature with the cross-ratio.
Comparison with affine curvature
Let us illustrate the preceding construction by comparison with geometri-
cally more transparent notion of the affine curvature and the affine param-
eter (see, e.g., [193]).](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-31-320.jpg)
![18 CHAPTER 1. INTRODUCTION
x x+ε
v
Figure 1.13: Cubic form on an affine curve
Consider a non-degenerate curve γ in the affine plane with a fixed area
form. We define a cubic form on γ as follows. Let v be a tangent vector to γ
at point x. Extend v to a tangent vector field along γ and denote by φt the
corresponding local one-parameter group of diffeomorphisms of γ. Consider
the segment between x and φε(x), see figure 1.13, and denote by A(x, v, ε)
the area, bounded by it and the curve. This function behaves cubically in
ε, and we define a cubic form
σ(x, v) = lim
ε→0
A(x, v, ε)
ε3
. (1.4.9)
A parameter t on γ is called affine if σ = c(dt)3 with a positive constant c.
By the very construction, the notion of affine parameter is invariant with
respect to the group of affine transformations of the plane while σ is invariant
under the (smaller) equiaffine group.
Alternatively, an affine parameter is characterized by the condition
|γ0
(t), γ00
(t)| = const.
Hence the vectors γ000(t) and γ0(t) are proportional: γ000(t) = −k(t)γ0(t). The
function k(t) is called the affine curvature.
The affine parameter is not defined at inflection points. The affine cur-
vature is constant if and only if γ is a conic.
Comment
The notion of projective curvature appeared in the literature in the second
half of the XIX-th century. From the very beginning, curves were studied
in the framework of differential operators – see [231] for an account of this
early period of projective differential geometry.
In his book [37], E. Cartan also calculated the projective curvature as
a function of the projective length parameter. However, he gave an inter-
pretation of the projective curvature in terms of a projective structure on
the curve. Cartan invented a geometrical construction of developing a non-
degenerate curve on its osculating conic. This construction is a projective](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-32-320.jpg)
![20 CHAPTER 1. INTRODUCTION
1) C(h) = 0 for all h ∈ H;
2) C(gh) = C(g) for all h ∈ H and g ∈ G;
3) C(hg) = Th (C(g)) for all h ∈ H and g ∈ G.
The property of a 1-cocycle to be H-relative is stronger than the condition
to be H-invariant.
Exercise 1.5.2. Check that the conditions 1) – 3) imply (1.5.3).
Tensor densities in dimension 1
All tensor fields on a one-dimensional manifold M are of the form:
φ = φ(x)(dx)λ
, (1.5.4)
where λ ∈ R and x is a local coordinate; φ is called a tensor density of
degree λ. The space of tensor densities is denoted by Fλ(M), or Fλ, for
short. Equivalently, a tensor density of degree λ is defined as a section of
the line bundle (T ∗M)⊗λ.
The group Diff(M) naturally acts on Fλ. To describe explicitly this
action, consider the space of functions C∞(M) and define a 1-parameter
family of Diff(M)-actions on this space:
Tλ
f−1 : φ(x) 7→ f0
λ
φ(f(x)) , f ∈ Diff(M) (1.5.5)
cf. formula (1.3.2) for quadratic differentials. The Diff(M)-module Fλ is
nothing else but the module (C∞(M), Tλ). Although all Fλ are isomorphic
to each other as vector spaces, Fλ and Fµ are not isomorphic as Diff(M)-
modules unless λ = µ (cf. [72]).
In the case M = S1, there is a Diff(M)-invariant pairing Fλ ⊗F1−λ → R
given by the integral
hφ(x)(dx)λ
, ψ(x)(dx)1−λ
i =
Z
S1
φ(x)ψ(x)dx
Example 1.5.3. In particular, F0 is the space of smooth functions, F1 is
the space of 1-forms, F2 is the space of quadratic differentials, familiar from
the definition of the Schwarzian derivative, while F−1 is the space of vector
fields. The whole family Fλ is of importance, especially for integer and
half-integer values of λ.
Exercise 1.5.4. a) Check that formula (1.5.5) indeed defines an action of
Diff(M), that is, for all diffeomorphisms f, g, one has Tλ
f ◦ Tλ
g = Tλ
f◦g.
b) Show that the Vect(M)-action on Fλ(M) is given by the formula
Lλ
h(x) d
dx
: φ(dx)λ
7→ (hφ0
+ λ h0
φ)(dx)λ
. (1.5.6)](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-34-320.jpg)
![1.5. SCHWARZIAN DERIVATIVE AS A COCYCLE OF DIFF(RP1
) 21
First cohomology with coefficients in tensor densities
Recall identity (1.3.4) for the Schwarzian derivative. This identity means
that the Schwarzian derivative defines a 1-cocycle f 7→ S(f −1) on Diff(RP1
)
with coefficients in the space of quadratic differentials F2(RP1
). This cocycle
is not a coboundary; indeed, unlike S(f), any coboundary (1.5.2) depends
only on the 1-jet of a diffeomorphism – see formula (1.5.5).
The Schwarzian derivative vanishes on the subgroup PGL(2, R), and thus
it is PGL(2, R)-invariant.
Let us describe the first cohomology of the group Diff(RP1
) with coeffi-
cients in Fλ. These cohomologies can be interpreted as equivalence classes
of affine modules (or extensions) on Fλ. If G is a Lie group and V its
module then a structure of affine module on V is a structure of G-module
on the space V ⊕ R defined by
e
Tg : (v, α) 7→ (Tg v + α C(g), α),
where C is a 1-cocycle on G with values in V . See Section 8.4 for more
information on affine modules and extensions.
Theorem 1.5.5. One has
H1
(Diff(RP1
); Fλ) =
(
R, λ = 0, 1, 2,
0, otherwise
(1.5.7)
We refer to [72] for details. The corresponding cohomology classes are
represented by the 1-cocycles
C0(f−1
) = ln f0
, C1(f−1
) =
f00
f0
dx, C2(f−1
) =
f000
f0
−
3
2
f00
f0
2
(dx)2
.
The first cocycle makes sense in Euclidean geometry and the second one
in affine geometry. Their restrictions to the subgroup PGL(2, R) are non-
trivial, hence these cohomology classes cannot be represented by PGL(2, R)-
relative cocycles.
One is usually interested in cohomology classes, not in the represent-
ing cocycles which, as a rule, depend on arbitrary choices. However, the
Schwarzian derivative is canonical in the following sense.
Theorem 1.5.6. The Schwarzian derivative is a unique (up to a constant)
PGL(2, R)-relative 1-cocycle on Diff(RP1
) with coefficients in F2.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-35-320.jpg)
![22 CHAPTER 1. INTRODUCTION
Proof. If there are two such cocycles, then, by Theorem 1.5.5, their linear
combination is a coboundary, and this coboundary vanishes on PGL(2, R).
Every coboundary is of the form
C(f) = Tf (φ) − φ
for some φ ∈ F2. Therefore one has a non-zero PGL(2, R)-invariant quadratic
differential. It remains to note that PGL(2, R) does not preserve any tensor
field on RP1
.
Exercise 1.5.7. Prove that the infinitesimal version of the Schwarzian
derivative is the following 1-cocycle on the Lie algebra Vect(RP1
):
h(x)
d
dx
7→ h000
(x) (dx)2
. (1.5.8)
1.6 Virasoro algebra: the coadjoint representation
The Virasoro algebra is one of the best known infinite-dimensional Lie al-
gebras, defined as a central extension of Vect(S1). A central extension of
a Lie algebra g is a Lie algebra structure on the space g ⊕ R given by the
commutator
[(X, α), (Y, β)] = ([X, Y ], c(X, Y )),
where X, Y ∈ g, α, β ∈ R and c : g → R is a 1-cocycle. The reader can find
more information on central extensions in Section 8.4.
Definition of the Virasoro algebra
The Lie algebra Vect(S1) has a central extension given by the so-called
Gelfand-Fuchs cocycle
c
h1(x)
d
dx
, h2(x)
d
dx
=
Z
S1
h0
1(x) h00
2(x) dx. (1.6.1)
The corresponding Lie algebra is called the Virasoro algebra and will be
denoted by Vir. This is a unique (up to isomorphism) non-trivial central
extension of Vect(S1) (cf. Lemma 8.5.3).
Exercise 1.6.1. Check the Jacobi identity for Vir.
Note that the cocycle (1.6.1) is obtained by pairing the cocycle (1.5.8)
with a vector field.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-36-320.jpg)
![1.6. VIRASORO ALGEBRA: THE COADJOINT REPRESENTATION23
Computing the coadjoint representation
To explain the relation of the Virasoro algebra to projective geometry we
use the notion of coadjoint representation defined as follows. A Lie algebra
g acts on its dual space by
had∗
X φ, Y i := −hφ, [X, Y ]i,
for φ ∈ g∗ and X, Y ∈ g. This coadjoint representation carries much infor-
mation about the Lie algebra.
The dual space to the Virasoro algebra is Vir∗
= Vect(S1)∗ ⊕ R. It is
always natural to begin the study of the dual space to a functional space
with its subspace called the regular dual. This subspace is spanned by the
distributions given by smooth compactly supported functions.
Consider the regular dual space, Vir∗
reg = C∞(S1)⊕R consisting of pairs
(u(x), c) where u(x) ∈ C∞(S1) and c ∈ R, so that
h(u(x), c), (h(x)d/dx, α)i :=
Z
S1
u(x)h(x)dx + cα.
The regular dual space is invariant under the coadjoint action.
Exercise 1.6.2. The explicit formula for the coadjoint action of the Vira-
soro algebra on its regular dual space is
ad∗
(hd/dx, α)(u, c) = (hu0
+ 2h0
u − c h000
, 0). (1.6.2)
Note that the center of Vir acts trivially.
A remarkable coincidence
In the first two terms of the above formula (1.6.2) we recognize the Lie
derivative (1.5.6) of quadratic differentials, the third term is nothing else
but the cocycle (1.5.8), so that the action (1.6.2) is an affine module (see
Section 1.5). Moreover, this action coincides with the natural Vect(S1)-
action on the space of Sturm-Liouville operators (for c = −1/2), see formula
(1.3.8).
Thus one identifies, as Vect(S1)-modules, the regular dual space Vir∗
reg
and the space of Sturm-Liouville operators
(u(x), c) ↔ −2c
d2
dx2
+ u(x) (1.6.3)
and obtains a nice geometrical interpretation for the coadjoint representation
of the Virasoro algebra.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-37-320.jpg)
![24 CHAPTER 1. INTRODUCTION
Remark 1.6.3. To simplify exposition, we omit the definition of the Vira-
soro group (the group analog of the Virasoro algebra) and the computation of
its coadjoint action which, indeed, coincides with the Diff(S1)-action (1.3.7).
Coadjoint orbits
The celebrated Kirillov’s orbit method concerns the study of the coadjoint
representation. Coadjoint orbits of a Lie algebra g are defined as integral
surfaces in g∗, tangent to the vector fields φ̇ = ad∗
Xφ for all X ∈ g 2.
Classification of the coadjoint orbits of a Lie group or a Lie algebra is always
an interesting problem.
The identification (1.6.3) makes it possible to express invariants of the
coadjoint orbits of the Virasoro algebra in terms of invariants of Sturm-
Liouville operators (and projective structures on S1, see Theorem 1.3.1).
An invariant of a differential operator on S1 is the monodromy operator
mentioned in Section 1.3. In the case of Sturm-Liouville operators, this is
an element of the universal covering ^
PGL(2, R).
Theorem 1.6.4. The monodromy operator is the unique invariant of the
coadjoint orbits of the Virasoro algebra.
Proof. Two elements (u0(x), c) and (u1(x), c) of Vir∗
reg belong to the same
coadjoint orbit if and only if there is a one-parameter family (ut(x), c) with
t ∈ [0, 1] such that, for every t, the element ( ˙
ut(x), 0) is the result of the
coadjoint action of Vir; here dot denotes the derivative with respect to t. In
other words, there exists ht(x) d
dx ∈ Vect(S1) such that
u̇t(x) = ht(x) u0
t(x) + 2h0
t(x) ut(x) − c h000
t (x). (1.6.4)
According to (1.6.3), a family (ut(x), c) defines a family of Sturm-Liouville
operators: Lt = −2c(d/dx)2 + u(x)t. Consider the corresponding family of
Sturm-Liouville equations
Lt(φ) = −2c φ00
(x) + u(x)t φ(x) = 0.
For every t, one has a two-dimensional space of solutions, hφ1t(x), φ2t(x)i.
Define a Vect(S1)-action on the space of solutions using the Leibnitz
rule:
(ad∗
h d
dx
L)(φ) + L(Th d
dx
φ) = 0
2
This definition allows us to avoid using the notion of a Lie group, and sometimes this
simplifies the situation, for instance, in the infinite-dimensional case.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-38-320.jpg)
![26 CHAPTER 1. INTRODUCTION
Remark 1.6.6. One can understand, in a more traditional way, the mon-
odromy operator as an element of SL(2, R), instead of its universal covering.
Then there is another discrete invariant, representing a class in π1(SL(2, R)).
This invariant is nothing but the winding number of the corresponding curve
in RP1
, see Section 1.3. For instance, there are infinitely many connected
components in the space of Sturm-Liouville operators with the same mon-
odromy.
Relation to infinite-dimensional symplectic geometry
A fundamental fact which makes the notion of coadjoint orbits so impor-
tant (in comparison with the adjoint orbits) is that every coadjoint orbit
has a canonical g-invariant symplectic structure (often called the Kirillov
symplectic form). Moreover, the space g∗ has a Poisson structure called
the Lie-Poisson(-Berezin-Kirillov-Kostant-Souriau) bracket, and the coad-
joint orbits are the corresponding symplectic leaves. See Section 8.2 for a
brief introduction to symplectic and Poisson geometry.
An immediate corollary of the above remarkable coincidence is that the
space of the Sturm-Liouville operators is endowed with a natural Diff(S1)-
invariant Poisson structure; furthermore, it follows from Theorem 1.6.4 that
the space of Sturm-Liouville operators with a fixed monodromy is an (infinite
dimensional) symplectic manifold.
Comment
The Virasoro algebra was discovered in 1967 by I. M. Gelfand and D. B.
Fuchs. It appeared in the physical literature around 1975 and became very
popular in conformal field theory (see [90] for a comprehensive reference).
The coadjoint representation of Lie groups and Lie algebras plays a spe-
cial role in symplectic geometry and representation theory, cf. [112]. The
observation relating the coadjoint representation to the natural Vect(S1)-
action on the space of Sturm-Liouville operators and, therefore, on the space
of projective structures on S1, and the classification of the coadjoint orbits
was made in 1980 independently by A. A. Kirillov and G. Segal [116, 186].
The classification of the coadjoint orbits then follows from the classical work
by Kuiper [126] (see also [130]) on classification of projective structures. Our
proof, using the homotopy method, is probably new.
This and other remarkable properties of the Virasoro algebra, its relation
with the Korteweg-de Vries equation, moduli spaces of holomorphic curves,
etc., make this infinite-dimensional Lie algebra one of the most interesting](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-40-320.jpg)
![2.1. INVARIANT DIFFERENTIAL OPERATORS ON RP1
31
Linear projectively invariant operators
Our goal is to describe differential operators on RP1
, invariant under pro-
jective transformations. In the one-dimensional case, there is only one type
of tensors, namely tensor densities φ(x)(dx)λ. Recall that the space of such
tensor densities is denoted by Fλ(RP1
).
A classical result of projective differential geometry is classification of
projectively invariant linear differential operators A : Fλ(RP1
) → Fµ(RP1
)
(see [28]).
Theorem 2.1.2. The space of PGL(2, R)-invariant linear differential oper-
ators on tensor densities is generated by the identity operator from Fλ(RP1
)
to Fλ(RP1
) and the operators of degree k given, in an affine coordinate, by
the formula
Dk : φ(x)(dx)
1−k
2 7→
dkφ(x)
dxk
(dx)
1+k
2 . (2.1.3)
Proof. The action of SL(2, R) is given, in an affine chart, by the formula
x 7→
ax + b
cx + d
. (2.1.4)
Exercise 2.1.3. Prove that the operators Dk are PGL(2, R)-invariant.
The infinitesimal version of formula 2.1.4 gives the action of the Lie
algebra sl(2, R).
Exercise 2.1.4. a) Prove that the sl(2, R)-action on RP1
is generated by
the three vector fields
d
dx
, x
d
dx
, x2 d
dx
. (2.1.5)
b) Prove that the corresponding action on Fλ(RP1
) is given by the following
operators (the Lie derivatives):
Lλ
d
dx
=
d
dx
, Lλ
x d
dx
= x
d
dx
+ λ, Lλ
x2 d
dx
= x2 d
dx
+ 2λ x . (2.1.6)
Consider now a differential operator
A = ak(x)
dk
dxk
+ · · · + a0(x)
from Fλ(RP1
) to Fµ(RP1
) and assume that A is SL(2, R)-invariant. This
means that
A ◦ Lλ
X = Lµ
X ◦ A](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-45-320.jpg)
![32 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE
for all X ∈ sl(2, R).
Take X = d/dx to conclude that all the coefficients ai(x) of A are con-
stants. Now take X = xd/dx:
Lµ
x d
dx
◦
X
ai
di
dxi
=
X
ai
di
dxi
◦ Lλ
x d
dx
.
Using (2.1.6), it follows that ai(i + λ − µ) = 0 for all i. Hence all ai but one
vanish, and µ = λ + k where k is the order of A.
Finally, take X = x2 d
dx . One has
Lλ+k
x2 d
dx
◦
dk
dxk
=
dk
dxk
◦ Lλ
x2 d
dx
.
If k ≥ 1 then, using (2.1.6) once again, one deduces that 2λ = 1 − k, as
claimed; if k = 0, then A is proportional to the identity.
The operator D1 is just the differential of a function. This is the only
operator invariant under the full group Diff(RP1
). The operator D2 is a
Sturm-Liouville operator already introduced in Section 1.3. Such an oper-
ator determines a projective structure on RP1
. Not surprisingly, the pro-
jective structure, corresponding to D2, is the standard projective structure
whose symmetry group is PGL(2, R). The geometric meaning of the opera-
tors Dk with k ≥ 3 will be discussed in the next section.
Comment
The classification problem of invariant differential operators was posed by
Veblen in his talk at ICM in 1928. Many important results have been ob-
tained since then. The only unitary invariant differential operator on tensor
fields (and tensor densities) is the de Rham differential, cf. [114, 179].
2.2 Curves in RPn
and linear differential operators
In Sections 1.3 and 1.4 we discussed the relations between non-degenerate
curves and linear differential operators in dimensions 1 and 2. In this section
we will extend this construction to the multi-dimensional case.
Constructing differential operators from curves
We associate a linear differential operator
A =
dn+1
dxn+1
+ an−1(x)
dn−1
dxn−1
+ · · · + a1(x)
d
dx
+ a0(x) (2.2.1)](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-46-320.jpg)
![34 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE
which is, of course, a particular case of the action of Diff(S1) on Dλ,µ(S1)
as in (2.1.1). In other words, A ∈ Dn+1
− n
2
, n+2
2
(S1).
Exercise 2.2.2. The action of Diff(S1) on Dλ,µ(S1) preserves the specific
form of the operators (2.2.1), namely, the highest-order coefficient equals 1
and the next highest equals zero, if and only if
λ = −
n
2
and µ =
n + 2
2
.
Theorem 2.2.3. The two Diff(S1)-actions on the space of differential op-
erators (2.2.1) coincide.
Proof. Let us start with the geometric action. Consider a new parameter
y = f(x) on γ. Then
Γx = Γyf0
, Γxx = Γyy(f0
)2
+ Γyf00
,
etc., where Γ is a lifted curve and f0 denotes df/dx. It follows that
|Γ, Γx, . . . , Γx...x| = |Γ, Γy, . . . , Γy...y|(f0
)n(n+1)/2
,
and, therefore, the Wronski determinant W(x) is a tensor density of degree
n(n + 1)/2, that is, an element of Fn(n+1)/2. Hence the coordinates of the
canonical lift αΓ given by (2.2.2) are tensor densities of degree −n/2 (we
already encountered a particular case n = 2 in Exercise 1.6.5). Being the
coordinates of the canonical lift αΓ, the solutions of the equation
Aφ = 0 (2.2.4)
are −n/2-densities.
From the very definition of the algebraic action (2.2.3) it follows that
the kernel of the operator A consists of −n/2-densities. It remains to note
that the kernel uniquely defines the corresponding operator.
The brevity of the proof might be misleading. An adventurous reader
may try to prove Theorem 2.2.3 by a direct computation. Even for the
Sturm-Liouville (n = 2) case this is quite a challenge (see, e.g. [37]).
Example 2.2.4. The SL(2, R)-invariant linear differential operator (2.1.3)
fits into the present framework. This operator corresponds to a remarkable
parameterized projective curve in RPk−1
, called the normal curve, uniquely
characterized by the following property. The parameter x belongs to S1 and](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-48-320.jpg)
![2.2. CURVES IN RPN
AND LINEAR DIFFERENTIAL OPERATORS37
for all parameter values x0.
Indeed, the vector Γ̃(i)(x0) belongs to the space of solutions U and this
solution vanishes at x0. The function b
Γ(x0) ∈ V vanishes at x0 with the
first n − 1 derivatives, and (2.2.9) follows from the above expression for the
operator B. Therefore γ̃∗ = b
γ, that is, the curves corresponding to A and
A∗ are projectively dual.
Exercise 2.2.7. Prove the following explicit formula:
B(φ, ψ) =
X
r+s+t≤n
(−1)r+t+1
r + t
r
a
(r)
r+s+t+1φ(s)
ψ(t)
.
Remark 2.2.8. If A is a symmetric operator, A∗ = A, then B is a non-
degenerate skew-symmetric bilinear form, i.e., a symplectic structure, on the
space Ker A – cf. [166].
Monodromy
If γ is a closed curve, then the operator A has periodic coefficients. The
converse is not at all true. Let A be an operator with periodic coefficients,
in other words, a differential operator on S1. The solutions of the equation
Aφ = 0 are not necessarily periodic; they are defined on R, viewed as the
universal covering of S1 = R/2πZ. One obtains a linear map on the space
of solutions:
T : φ(x) 7→ φ(x + 2π)
called the monodromy. Monodromy was already mentioned in Sections 1.3
and 1.6.
Consider in more detail the case of operators (2.2.1). The Wronski de-
terminant of any (n+1)-tuple of solutions is constant. This defines a volume
form on the space of solutions. Since T preserves the Wronski determinant,
the monodromy belongs to SL(n + 1, R). Note however that this element
of SL(n + 1, R) is defined up to a conjugation, for there is no natural basis
in ker A and there is no way to identify ker A with Rn+1; only a conjugacy
class of T has an invariant meaning.
Consider a projective curve γ(x) associated with a differential operator
A on S1. Let Γ(x) ⊂ Rn+1 be the canonical lift of γ(x). Both curves are
not necessarily closed, but satisfy the monodromy condition
γ(x + 2π) = T(γ(x)), Γ(x + 2π) = T(Γ(x)),
where T is a representative of a conjugacy class in SL(n + 1, R).](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-51-320.jpg)
![38 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE
As a consequence of Theorem 2.2.3, asserting the coincidence of the
algebraic Diff(S1)-action (2.2.3) on the space of differential operators with
the geometric action by reparameterization, we have the following statement.
Corollary 2.2.9. The conjugacy class in SL(n + 1, R) of the monodromy
of a differential operator (2.2.1) is invariant with respect to the Diff+(S1)-
action, where Diff+(S1) is the connected component of Diff(S1).
Comment
Representation of parameterized non-degenerate curves in RPn
(modulo
equivalence) by linear differential operators was a basic idea of projective
differential geometry of the second half of XIX-th century. We refer to
Wilczynski’s book [231] for a first systematic account of this approach. Our
proof of Theorem 2.2.6 follows that of [231] and [13]; a different proof can
be found in [106].
2.3 Homotopy classes of non-degenerate curves
Differential operators on RP1
of the special form (2.2.1) correspond to non-
degenerate curves in RPn
. In this section we give a topological classi-
fication of such curves. We study homotopy equivalence classes of non-
degenerate immersed curves with respect to the homotopy, preserving the
non-degeneracy. This allows us to distinguish interesting classes of curves,
such as that of convex curves.
Curves in S2
: a theorem of J. Little
Let us start with the simplest case, the classification problem for non-
degenerate curves on the 2-sphere.
Figure 2.1: Non-degenerate curves on S2
Theorem 2.3.1. There are 3 homotopy classes of non-degenerate immersed
non-oriented closed curves on S2 represented by the curves in figure 2.1.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-52-320.jpg)
![40 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE
Proof. Arguing by contradiction, assume that there is a homotopy destroy-
ing convexity. Convexity is an open condition. Consider the first moment
when the curve fails to be convex. At this moment, there exists a great cir-
cle intersecting the curve with total multiplicity four. The following 3 cases
are possible: a) four distinct transverse intersections, b) two transverse and
one tangency, c) two tangencies, see figure 2.3. Note that a non-degenerate
curve cannot have intersection multiplicity 2 with a great circle at a point.
a)
b) c)
Figure 2.3: Total multiplicity 4
Case a) is impossible: since transverse intersection is an open condition,
this cannot be the first moment when the curve fails to be convex. In cases
b) and c) one can perturb the great circle so that the intersections become
transverse and we are back to case a). In case b) this is obvious, as well as
in case c) if the two points of tangency are not antipodal, see figure 2.3. For
antipodal points, one rotates the great circle about the axis connecting the
tangency points.
Part 2. Let us now prove that a non-degenerate curve on S2 is non-
degenerate homotopic to a curve in figure 2.1. Unlike the planar case, non-
degenerate curves with winding number n and n + 2, where n ≥ 2, are
non-degenerate homotopic, see figure 2.4. The apparent inflection points
are not really there; see [137] for a motion picture featuring front-and-back
view. The authors recommend the reader to repeat their experience and to
draw the picture on a well inflated ball.
Therefore, any curve in figure 2.2 is, indeed, homotopic to a curve in
figure 2.1 in the class of non-degenerate curves.
Lemma 2.3.4. A non-degenerate curve on the 2-sphere is homotopic, in
the class of non-degenerate curves, to a curve that lies in a hemisphere.
Proof. If the curve is convex, then it already lies in a hemisphere. The proof
of this fact is similar to that of Lemma 2.3.3. If the curve is not convex, then](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-54-320.jpg)
![knight in Marmion, who
On high his forky pennon bore,
Like swallow's tail in shape and hue.
We read in the Roll of Karlaverok, as early as the year 1300, of
Many a beautiful pennon fixed to a lance,
And many a banner displayed;
and of the knight in Chaucer's Canterbury Tales, we hear that
By hys bannere borne is hys pennon
Of golde full riche.
The pennon bore the arms of the knight, and they were in the
earlier days of chivalry so emblazoned upon it as to appear in their
proper position not when the lance was held erect but when held
horizontally for the charge. The earliest brass now extant, that of Sir
John Daubernoun, at Stoke d'Abernon Church, in Surrey, represents
the knight as bearing a lance with pennon. Its date is 1277, and the
device is a golden chevron on a field of azure. In this example the
pennon, instead of being forked, comes to a single point.
The pennon was the ensign of those knights who were not
bannerets, and the bearers of it were therefore sometimes called
pennonciers; the term is derived from the Latin word for a feather,
penna, from the narrow, elongated form. The pennons of our lancer
regiments (Fig. 30) give one a good idea of the form, size, and
general effect of the ancient knightly pennon, though they do not
bear distinctive charges upon them, and thus fail in one notable
essential to recall to our minds the brilliant blazonry and variety of
device that must have been so marked and effective a feature when
the knights of old took the field. In a drawing of the year 1813, of
the Royal Horse Artillery, we find the men armed with lances, and
these with pennons of blue and white, as we see in Fig. 31.[11]](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-57-320.jpg)
![Of the thirty-seven pennons borne on lances by various knights
represented in the Bayeux tapestry, twenty-eight have triple points,
while others have two, four, or five. The devices upon these pennons
are very various and distinctive, though the date is before the period
of the definite establishment of heraldry. Examples of these may be
seen in Figs. 39, 40, 41, 42.
The pennoncelle, or pencel, is a diminutive of the pennon, small as
that itself is. Such flags were often supplied in large quantities at
any special time of rejoicing or of mourning. At the burial in the year
1554 of the nobull Duke of Norffok, we note amongst other items
a dosen of banerolles of ys progene, a standard, a baner of
damaske, and xij dosen penselles. At the burial of Sir William
Goring we find ther was viij dosen of penselles, while at the Lord
Mayor's procession in 1555 we read that there were ij goodly
pennes [State barges] deckt with flages and stremers and a m
penselles. This m, or thousand, we can perhaps scarcely take
literally, though in another instance we find the cordes were hanged
with innumerable pencelles.[12]
The statement of the cost of the funeral of Oliver Cromwell is
interesting, as we see therein the divers kinds of flags that graced
the ceremony. The total cost of the affair was over £28,000, and the
unhappy undertaker, a Mr. Rolt, was paid very little, if any, of his bill.
The items include six gret banners wrought on rich taffaty in oil,
and gilt with fine gold, at £6 each. Five large standards, similarly
wrought, at a cost of £10 each; six dozen pennons, a yard long, at a
sovereign each; forty trumpet banners, at forty shillings apiece;
thirty dozen of pennoncelles, a foot long, at twenty shillings a dozen;
and twenty dozen ditto at twelve shillings the dozen. Poor Rolt!
In the accompte and reckonyng for the Lord Mayor's Show of 1617
we find payde to Jacob Challoner, painter, for a greate square
banner, the Prince's Armes, the somme of seven pounds. We also
find, More to him for the new payntyng and guyldyng of ten
trumpet banners, for payntyng and guyldyng of two long pennons of](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-58-320.jpg)
![badge of the Warwicks,[13] so we find it at the head adorned with a
grete bere holding a ragged staffe, and the rest of its length
powdrid full of raggid staves,
A stately ship,
With all her bravery on, and tackle trim,
Sails filled, and streamers waving.
Machyn tells us in his diary for August 3rd, 1553, how The Queen
came riding to London, and so on to the Tower, makyng her entry at
Aldgate, and a grett nombur of stremars hanging about the sayd
gate, and all the strett unto Leydenhalle and unto the Tower were
layd with graffel, and all the crafts of London stood with their banars
and stremars hangyd over their heds. In the picture by Volpe in the
collection at Hampton Court of the Embarkation of Henry VIII. from
Dover in the year 1520, to meet Francis I. at the Field of the Cloth of
Gold, we find, very naturally, a great variety and display of flags of
all kinds. Figs. 20, 21, 23 are streamers therein depicted, the
portcullis, Tudor rose, and fleur-de-lys being devices of the English
king, while the particular ground upon which they are displayed is in
each case made up of green and white, the Tudor livery colours. We
may see these again in Fig. 71, where the national flag of the Cross
of St. George has its white field barred with the Tudor green. In the
year 1554 even the naval uniform of England was white and green,
both for officers and mariners, and the City trained bands had white
coats welted with green. Queen Elizabeth, though of the Tudor race,
took scarlet and black as her livery colours; the House of
Plantaganet white and red; of York, murrey and blue; of Lancaster,
white and blue; of Stuart, red and yellow. The great nobles each
also had their special liveries; thus in a grand review of troops on
Blackheath, on May 16th, 1552, we find that the Yerle of Pembroke
and ys men of armes had cotes blake bordered with whyt, while
the retainers of the Lord Chamberlain were in red and white, those
of the Earl of Huntingdon in blue, and so forth.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-60-320.jpg)
![same colours are introduced, but with due regard to heraldic law,
and certainly with far more pleasing effect.
One sees the same obedience to this rule in the special flags used
for signalling, where great clearness of definition at considerable
distances is an essential. Such combinations as blue and black, red
and blue, yellow and white, carry their own condemnation with
them, as anyone may test by actual experiment; stripes of red and
blue, for instance, at a little distance blending into purple, while
white and yellow are too much alike in strength, and when the
yellow has become a little faded and the white a little dingy they
appear almost identical. We have this latter combination in Fig. 198,
the flag of the now vanished Papal States. It is a very uncommon
juxtaposition, and only occurs in this case from a special religious
symbolism into which we need not here enter. The alternate red and
green stripes in Fig. 63 are another violation of the rule, and have a
very confusing effect.[14]
The colours of by far the greatest frequency of occurrence are red,
white, and blue; yellow also is not uncommon; orange is only found
once, in Fig. 249, where it has a special significance, since this is the
flag of the Orange Free State. Green occurs sparingly. Italy (Fig.
197) is perhaps the best known example. We also find it in the
Brazilian flag (Fig. 169), the Mexican (Fig. 172), in the Hungarian
tricolor (Fig. 214), and in Figs. 199, 201, 209, the flags of smaller
German States, but it is more especially associated with
Mohammedan States, as in Figs. 58, 63, 64, 235. Black is found but
seldom, but as heraldic requirements necessitate that it should be
combined either with white or yellow, it is, when seen, exceptionally
brilliant and effective. We see it, for example, in the Royal Standard
of Spain, (Fig. 194), in Figs. 207 and 208, flags of the German
Empire, in Fig. 226, the Imperial Standard of Russia, and in Fig. 236,
the brilliant tricolor of the Belgians.[15]
In orthodox flags anything of the nature of an inscription is very
seldom seen. We find a reference to order and progress on the](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-63-320.jpg)
![Brazilian flag (Fig. 169), while the Turkish Imperial Standard (Fig.
238) bears on its scarlet folds the monogram of the Sultan; but
these exceptions are rare.[16] We have seen that, on the contrary,
on the flags of insurgents and malcontents the inscription often
counts for much. On the alteration of the style in the year 1752 this
necessary change was made the subject of much ignorant reproach
of the government of the day, and was used as a weapon of party
warfare. An amusing instance of this feeling occurs in the first plate
of Hogarth's election series, where a malcontent, or perhaps only a
man anxious to earn a shilling, carries a big flag inscribed, Give us
back our eleven days. The flags of the Covenanters often bore
mottoes or texts. Fig. 34 is a curious example: the flag hoisted by
the crew of H.M.S. Niger when they opposed the mutineers in 1797
at Sheerness. It is preserved in the Royal United Service Museum. It
is, as we have seen, ordinarily the insubordinate and rebellious who
break out into inscriptions of more or less piety or pungency, but we
may conclude that the loyal sailors fighting under the royal flag
adopted this device in addition as one means the more of fighting
the rebels with their own weapons.
During the Civil War between the Royalists and Parliamentarians, we
find a great use made of flags inscribed with mottoes. Thus, on one
we see five hands stretching at a crown defended by an armed hand
issuing from a cloud, and the motto, Reddite Cæsari. In another
we see an angel with a flaming sword treading a dragon underfoot,
and the motto, Quis ut Deus, while yet another is inscribed,
Courage pour la Cause. On a fourth we find an ermine, and the
motto, Malo mori quam fœdari—It is better to die than to be
sullied, in allusion to the old belief that the ermine would die rather
than soil its fur. Hence it is the emblem of purity and stainless
honour.
The blood-red flag is the symbol of mutiny and of revolution. As a
sign of disaffection it was twice, at the end of last century, displayed
in the Royal Navy. A mutiny broke out at Portsmouth in April, 1797,
for an advance of pay; an Act of Parliament was passed to sanction](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-64-320.jpg)
![little book published in 1848, many of the flags are at once seen to
be now out of date. The particular year was one of exceptional
political agitation, and the author evidently felt that his work was
almost old-fashioned even on its issue. The accompanying
illustrations, he says, having been completed prior to the recent
revolutionary movements on the Continent of Europe, it has been
deemed expedient to issue the plate in its present state, rather than
adopt the various tri-coloured flags, which cannot be regarded as
permanently established in the present unsettled state of political
affairs. The Russian American Company's flag, Fig. 59, that of the
States of the Church, of the Kingdom of Sardinia, the Turkish
Imperial Standard, Fig. 64, and many others that he gives, are all
now superseded. For Venice he gives two flags, that for war and that
for the merchant service. In each case the flag is scarlet, having a
broad band of blue, which we may take to typify the sea, near its
lower edge. From this rises in gold the winged lion of St. Mark,
having in the war ensign a sword in his right paw, and in the
peaceful colours of commerce a cross. Of thirty-five flags of all
nations, given as a supplement to the Illustrated London News in
1858, we note that eleven are now obsolete: the East India
Company, for instance, being now extinct, the Ionian Islands ceded
to Greece, Tuscany and Naples absorbed into Italy, and so forth.
In Figs. 52 and 53 we have examples of early Spanish flags, and in
54 and 55 of Portuguese, each and all being taken from a very
quaint map of the year 1502. This map may be said to be practically
the countries lying round the Atlantic Ocean, giving a good slice of
Africa, a portion of the Mediterranean basin, the British Isles, most
of South America, a little of North America, the West Indies,[17] etc.,
the object of the map being to show the division that Pope
Alexander VI. kindly made between those faithful daughters of the
Church—Spain and Portugal—of all the unclaimed portions of the
world. Figs. 52 and 53 are types of flags flying on various Spanish
possessions, while Figs. 54 and 55 are placed at different points on
the map where Portugal held sway. On one place in Africa we see
that No. 54 is surmounted by a white flag bearing the Cross of St.](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-67-320.jpg)
![Foremost amongst the flags of the British Empire the Royal Standard
takes its position as the symbol of the tie that unites all into one
great State. Its glowing blazonry of blue and scarlet and gold is
brought before us in Fig. 44. The three golden lions on the scarlet
ground are the device of England, the golden harp on the azure field
is the device of Ireland, while the ruddy lion rampant on the field of
gold[18] stands for Scotland. It may perhaps appear to some of our
readers that the standard of the Empire should not be confined to
such narrow limits; that the great Dominion of Canada, India,
Australia, the ever-growing South Africa, might justly claim a place.
Precedent, too, might be urged, since in previous reigns, Nassau,
Hanover, and other States have found a resting-place in its folds,
and there is much to be said in favour of a wider representation of
the greater component parts of our world-wide Empire; but two
great practical difficulties arise: the first is that the grand simplicity
of the flag would be lost if eight or ten different devices were
substituted for the three; and secondly, it would very possibly give
rise to a good deal of jealousy and ill-feeling, since it would be
impossible to introduce all. As it at present stands, it represents the
central home of the Empire, the little historic seed-plot from whence
all else has sprung, and to which all turn their eyes as the centre of
the national life. All equally agree to venerate the dear mother land,
but it is perhaps a little too much to expect that the people of
Jamaica or Hong Kong would feel the same veneration for the
beaver and maple-leaves of Canada, the golden Sun of India, or the
Southern Cross of Australasia. As it must clearly be all or none, it
seems that only one solution of the problem, the present one, is
possible. In the same way the Union flag (Fig. 90) is literally but the
symbol of England, Scotland, and Ireland, but far and away outside
its primary significance, it floats on every sea the emblem of that
Greater Britain in which all its sons have equal pride, and where all
share equal honour as brethren of one family.
The earliest Royal Standard bore but the three lions of England, and
we shall see presently that in different reigns various modifications
of its blazonry arose, either the result of conquest or of dynastic](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-70-320.jpg)
![possessions. Thus Figs. 43 and 44, though they bear a superficial
likeness, tell a very different story; the first of these, that of George
III., laying claim in its fourth quartering to lordship over Hanover
and other German States, and in its second quarter to the entirely
shadowy and obsolete claim over France, as typified by the golden
fleurs-de-lys on the field of azure.
How the three lions of England arose is by no means clear. Two lions
were assigned as the arms of William the Conqueror, but there is no
real evidence that he bore them. Heraldry had not then become a
definite science, and when it did a custom sprang up of assigning to
those who lived and died before its birth certain arms, the kindly
theory being that such persons, had they been then living, would
undoubtedly have borne arms, and that it was hard, therefore, that
the mere accident of being born a hundred years too soon should
debar them from possessing such recognition of their rank. Even so
late as Henry II. the bearing is still traditional, and it is said that on
his marriage with Alianore, eldest daughter of William, Duke of
Aquitaine and Guienne, he incorporated with his own two lions the
single lion that (it is asserted) was the device of his father-in-law. All
this, however, is theory and surmise, and we do not really find
ourselves on the solid ground of fact until we come to the reign of
Richard Cœur-de-Lion. Upon his second Great Seal we have the
three lions just as they are represented in Figs. 22, 43, 44, and as
they have been borne for centuries by successive sovereigns on their
arms, standards, and coinage, and as our readers may see them this
day on the Royal Standard and on much of the money they may take
out of their pockets. The date of this Great Seal of King Richard is
1195 A.D., so we have, at all events, a period of over seven hundred
years, waiving a break during the Commonwealth, in which the three
golden lions on their scarlet field have typified the might of England.
The rampant lion within the tressure, the device of Scotland—seen
in the second quarter of our Royal Standard, Fig. 44—is first seen on
the Great Seal of King Alexander II., about A.D. 1230, and the same
device, without any modification of colour or form[19] was borne by](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-71-320.jpg)
![gold upon it, and James I. adopted this, but without the crown, as a
quartering in his standard: its first appearance on the Royal
Standard of England.
Why Henry VIII. substituted the harp for the three crowns is not
really known. Some would have us believe that the king was
apprehensive that the three crowns might be taken as symbolising
the triple crown of the Pope; while others suggest that Henry, being
presented by the Pope with the supposed harp of Brian Boru, was
induced to change the arms of Ireland by placing on her coins the
representation of this relic of her most celebrated native king. The
Earl of Northampton, writing in the reign of James I., suggests yet a
third explanation. The best reason, saith he, that I can observe
for the bearing thereof is, it resembles that country in being such an
instrument that it requires more cost to keep it in tune than it is
worth.[20]
The Royal Standard should only be hoisted when the Sovereign or
some member of the royal family is actually within the palace or
castle, or at the saluting point, or on board the vessel where we see
it flying, though this rule is by no means observed in practice. The
only exception really permitted to this is that on certain royal
anniversaries it is hoisted at some few fortresses at home and
abroad that are specified in the Queen's Regulations.
The Royal Standard of England was, we have seen, in its earliest
form a scarlet flag, having three golden lions upon it, and it was so
borne by Richard I., John, Henry III., Edward I., and Edward II.
Edward III. also bore it for the first thirteen years of his reign, so
that this simple but beautiful flag was the royal banner for over one
hundred and fifty years. Edward III., on his claim in the year 1340 to
be King of France as well as of England, quartered the golden fleurs-
de-lys of that kingdom with the lions of England.[21] This remained
the Royal Standard throughout the rest of his long reign. Throughout
the reign of Richard II. (1377 to 1399) the royal banner was divided
in half by an upright line, all on the outer half being like that of](https://image.slidesharecdn.com/444591-250518161525-9edc96cd/85/Projective-Differential-Geometry-Old-And-New-Web-Version-V-Ovsienko-75-320.jpg)
Projective Differential Geometry Old And New Web Version V Ovsienko
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- 5. Projective differential geometry old and new: from Schwarzian derivative to cohomology of diffeomorphism groups V. Ovsienko1 S. Tabachnikov2 1 CNRS, Institut Girard Desargues Université Claude Bernard Lyon 1, 21 Avenue Claude Bernard, 69622 Villeurbanne Cedex, FRANCE; ovsienko@igd.univ-lyon1.fr 2 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA; tabachni@math.psu.edu
- 6. ii
- 7. Contents Preface: why projective ? vii 1 Introduction 1 1.1 Projective space and projective duality . . . . . . . . . . . . . 1 1.2 Discrete invariants and configurations . . . . . . . . . . . . . 5 1.3 Introducing Schwarzian derivative . . . . . . . . . . . . . . . 9 1.4 Further example of differential invariants: projective curvature 14 1.5 Schwarzian derivative as a cocycle of Diff(RP1 ) . . . . . . . . 19 1.6 Virasoro algebra: the coadjoint representation . . . . . . . . . 22 2 Geometry of projective line 29 2.1 Invariant differential operators on RP1 . . . . . . . . . . . . . 29 2.2 Curves in RPn and linear differential operators . . . . . . . . 32 2.3 Homotopy classes of non-degenerate curves . . . . . . . . . . 38 2.4 Two differential invariants of curves: projective curvature and cubic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.5 Projectively equivariant symbol calculus . . . . . . . . . . . . 45 3 Algebra of projective line and cohomology of Diff(S1) 51 3.1 Transvectants . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 First cohomology of Diff(S1) with coefficients in differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Application: geometry of differential operators on RP1 . . . . 61 3.4 Algebra of tensor densities on S1 . . . . . . . . . . . . . . . . 66 3.5 Extensions of Vect(S1) by the modules Fλ(S1) . . . . . . . . 70 4 Vertices of projective curves 75 4.1 Classic 4-vertex and 6-vertex theorems . . . . . . . . . . . . . 75 4.2 Ghys’ theorem on zeroes of the Schwarzian derivative and geometry of Lorentzian curves . . . . . . . . . . . . . . . . . . 82 iii
- 8. iv CONTENTS 4.3 Barner theorem on inflections of projective curves . . . . . . . 86 4.4 Applications of strictly convex curves . . . . . . . . . . . . . . 91 4.5 Discretization: geometry of polygons, back to configurations . 96 4.6 Inflections of Legendrian curves and singularities of wave fronts102 5 Projective invariants of submanifolds 109 5.1 Surfaces in RP3 : differential invariants and local geometry . . 110 5.2 Relative, affine and projective differential geometry of hyper- surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.3 Geometry of relative normals and exact transverse line fields 129 5.4 Complete integrability of the geodesic flow on the ellipsoid and of the billiard map inside the ellipsoid . . . . . . . . . . . 140 5.5 Hilbert’s 4-th problem . . . . . . . . . . . . . . . . . . . . . . 147 5.6 Global results on surfaces . . . . . . . . . . . . . . . . . . . . 154 6 Projective structures on smooth manifolds 159 6.1 Definition, examples and main properties . . . . . . . . . . . 160 6.2 Projective structures in terms of differential forms . . . . . . 165 6.3 Tensor densities and two invariant differential operators . . . 168 6.4 Projective structures and tensor densities . . . . . . . . . . . 170 6.5 Moduli space of projective structures in dimension 2, by V. Fock and A. Goncharov . . . . . . . . . . . . . . . . . . . . . 176 7 Multi-dimensional Schwarzian derivatives and differential operators 187 7.1 Multi-dimensional Schwarzian with coefficients in (2, 1)-tensors188 7.2 Projectively equivariant symbol calculus in any dimension . . 193 7.3 Multi-dimensional Schwarzian as a differential operator . . . 199 7.4 Application: classification of modules D2 λ(M) for an arbitrary manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.5 Poisson algebra of tensor densities on a contact manifold . . . 205 7.6 Lagrange Schwarzian derivative . . . . . . . . . . . . . . . . . 213 8 Appendices 223 8.1 Five proofs of the Sturm theorem . . . . . . . . . . . . . . . . 223 8.2 Language of symplectic and contact geometry . . . . . . . . . 226 8.3 Language of connections . . . . . . . . . . . . . . . . . . . . . 232 8.4 Language of homological algebra . . . . . . . . . . . . . . . . 235 8.5 Remarkable cocycles on groups of diffeomorphisms . . . . . . 238 8.6 Godbillon-Vey class . . . . . . . . . . . . . . . . . . . . . . . . 242
- 9. 8.7 Adler-Gelfand-Dickey bracket and infinite-dimensional Pois- son geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Bibliography 251 Index 268
- 10. vi CONTENTS
- 11. Preface: why projective ? Metrical geometry is a part of descriptive geometry1, and de- scriptive geometry is all geometry. Arthur Cayley On October 5-th 2001, the authors of this book typed in the word “Schwarzian” in the MathSciNet database and the system returned 666 hits. Every working mathematician has encountered the Schwarzian derivative at some point of his education and, most likely, tried to forget this rather scary expression right away. One of the goals of this book is to convince the reader that the Schwarzian derivative is neither complicated nor exotic, in fact, this is a beautiful and natural geometrical object. The Schwarzian derivative was discovered by Lagrange: “According to a communication for which I am indebted to Herr Schwarz, this expression occurs in Lagrange’s researches on conformable representation ‘Sur la con- struction des cartes géographiques’ ” [117]; the Schwarzian also appeared in a paper by Kummer in 1836, and it was named after Schwarz by Cayley. The main two sources of current publications involving this notion are classical complex analysis and one-dimensional dynamics. In modern mathematical physics, the Schwarzian derivative is mostly associated with conformal field theory. It also remains a source of inspiration for geometers. The Schwarzian derivative is the simplest projective differential invari- ant, namely, an invariant of a real projective line diffeomorphism under the natural SL(2, R)-action on RP1 . The unavoidable complexity of the for- mula for the Schwarzian is due to the fact that SL(2, R) is so large a group (three-dimensional symmetry group of a one-dimensional space). Projective geometry is simpler than affine or Euclidean ones: in pro- jective geometry, there are no parallel lines or right angles, and all non- degenerate conics are equivalent. This shortage of projective invariants is 1 By descriptive geometry Cayley means projective geometry, this term was in use in mid-XIX-th century. vii
- 12. viii PREFACE: WHY PROJECTIVE ? due to the fact that the group of symmetries of the projective space RPn is large. This group, PGL(n+1, R), is equal to the quotient of GL(n+1, R) by its center. The greater the symmetry group, the fewer invariants it has. For instance, there exists no PGL(n + 1, R)-invariant tensor field on RPn , such as a metric or a differential form. Nevertheless, many projective invariants have been found, from Ancient Greeks’ discovery of configuration theorems to differential invariants. The group PGL(n + 1, R) is maximal among Lie groups that can act effectively on n-dimensional manifolds. It is due to this maximality that projective differential invariants, such as the Schwarzian derivative, are uniquely determined by their invariance properties. Once projective geometry used to be a core subject in university curricu- lum and, as late as the first half of the XX-th century, projective differential geometry was a cutting edge geometric research. Nowadays this subject occupies a more modest position, and a rare mathematics major would be familiar with the Pappus or Desargues theorems. This book is not an exhaustive introduction to projective differential geometry or a survey of its recent developments. It is addressed to the reader who wishes to cover a greater distance in a short time and arrive at the front line of contemporary research. This book can serve as a basis for graduate topics courses. Exercises play a prominent role while historical and cultural comments relate the subject to a broader mathematical context. Parts of this book have been used for topic courses and expository lectures for undergraduate and graduate students in France, Russia and the USA. Ideas of projective geometry keep reappearing in seemingly unrelated fields of mathematics. The authors of this book believe that projective differential geometry is still very much alive and has a wealth of ideas to offer. Our main goal is to describe connections of the classical projective geometry with contemporary research and thus to emphasize unity of mathematics. Acknowledgments. For many years we have been inspired by our teachers V. I. Arnold, D. B. Fuchs and A. A. Kirillov who made a significant contribution to the modern understanding of the material of this book. It is a pleasure to thank our friends and collaborators C. Duval, B. Khesin, P. Lecomte and C. Roger whose many results are included here. We are much indebted to J. C. Alvarez, M. Ghomi, E. Ghys, J. Landsberg, S. Parmentier, B. Solomon, G. Thorbergsson and M. Umehara for enlightening discussions and help. It was equally pleasant and instructive to work with our younger colleagues and students S. Bouarroudj, H. Gargoubi, L. Guieu and S. Morier- Genoud. We are grateful to the Shapiro Fund at Penn State, the Research in Pairs program at Oberwolfach and the National Science Foundation for
- 14. x PREFACE: WHY PROJECTIVE ?
- 15. Chapter 1 Introduction ...the field of projective differential geometry is so rich that it seems well worth while to cultivate it with greater energy than has been done heretofore. E. J. Wilczynski In this introductory chapter we present a panorama of the subject of this book. The reader who decides to restrict himself to this chapter will get a rather comprehensive impression of the area. We start with the classical notions of curves in projective space and de- fine projective duality. We then introduce first differential invariants such as projective curvature and projective length of non-degenerate plane pro- jective curves. Linear differential operators in one variable naturally appear here to play a crucial role in the sequel. Already in the one-dimensional case, projective differential geometry of- fers a wealth of interesting structures and leads us directly to the celebrated Virasoro algebra. The Schwarzian derivative is the main character here. We tried to present classical and contemporary results in a unified synthetic manner and reached the material discovered as late as the last decades of the XX-th century. 1.1 Projective space and projective duality Given a vector space V , the associated projective space, P(V ), consists of one-dimensional subspaces of V . If V = Rn+1 then P(V ) is denoted by RPn . The projectivization, P(U), of a subspace U ⊂ V is called a projective subspace of P(V ). 1
- 16. 2 CHAPTER 1. INTRODUCTION The dual projective space P(V )∗ is the projectivization of the dual vec- tor space V ∗. Projective duality is a correspondence between projective subspaces of P(V ) and P(V )∗, the respective linear subspaces of V and V ∗ are annulators of each other. Note that projective duality reverses the inci- dence relation. Natural local coordinates on RPn come from the vector space Rn+1. If x0, x1, . . . , xn are linear coordinates in Rn+1, then yi = xi/x0 are called affine coordinates on RPn ; these coordinates are defined in the chart x0 6= 0. Likewise, one defines affine charts xi 6= 0. The transition functions between two affine coordinate systems are fractional-linear. Projectively dual curves in dimension 2 The projective duality extends to curves. A smooth curve γ in RP2 deter- mines a 1-parameter family of its tangent lines. Each of these lines gives a point in the dual plane RP2∗ and we obtain a new curve γ∗ in RP2∗ , called the dual curve. In a generic point of γ, the dual curve is smooth. Points in which γ∗ has singularities correspond to inflection of γ. In generic points, γ has order 1 contact with its tangent line; inflection points are those points where the order of contact is higher. γ γ∗ Figure 1.1: Duality between an inflection and a cusp Exercise 1.1.1. a) Two parabolas, given in affine coordinates by y = xα and y = xβ, are dual for 1/α + 1/β = 1. b) The curves in figure 1.2 are dual to each other. A fundamental fact is that (γ∗)∗ = γ which justifies the terminology (a proof given in the next subsection). As a consequence, one has an alternative definition of the dual curve. Every point of γ determines a line in the dual plane, and the envelope of these lines is γ∗. Two remarks are in order. The definition of the dual curve extends to curves with cusps, provided the tangent line is defined at every point and
- 17. 1.1. PROJECTIVE SPACE AND PROJECTIVE DUALITY 3 γ γ∗ Figure 1.2: Projectively dual curves depends on the point continuously. Secondly, duality interchanges double points with double tangent lines. Exercise 1.1.2. Consider a generic smooth closed immersed plane curve γ. Let T± be the number of double tangent lines to γ such that locally γ lies on one side (respectively, opposite sides) of the double tangent, see figure 1.3, I the number of inflection points and N the number of double points of γ. Prove that T+ − T− − 1 2 I = N. T+ T- I N Figure 1.3: Invariants of plane curves Hint. Orient γ and let `(x) be the positive tangent ray at x ∈ γ. Consider the number of intersection points of `(x) with γ and investigate how this number changes as x traverses γ. Do the same with the negative tangent ray. Projective curves in higher dimensions Consider a generic smooth parameterized curve γ(t) in RPn and its generic point γ(0). Construct a flag of subspaces as follows. Fix an affine coor-
- 18. 4 CHAPTER 1. INTRODUCTION dinate system, and define the k-th osculating subspace Fk as the span of γ0(0), γ00(0), . . . , γ(k)(0). This projective space depends neither on the pa- rameterization nor on the choice of affine coordinates. For instance, the first osculating space is the tangent line; the n−1-th is called the osculating hyperplane. A curve γ is called non-degenerate if, in every point of γ, one has the full osculating flag F1 ⊂ · · · ⊂ Fn = RPn . (1.1.1) A non-degenerate curve γ determines a 1-parameter family of its osculating hyperplanes. Each of these hyperplanes gives a point in the dual space RPn∗ , and we obtain a new curve γ∗ called the dual curve. As before, one has the next result. Theorem 1.1.3. The curve, dual to a non-degenerate one, is smooth and non-degenerate, and (γ∗)∗ = γ. Proof. Let γ(t) be a non-degenerate parameterized curve in RPn , and Γ(t) its arbitrary lift to Rn+1. The curve γ∗(t) lifts to a curve Γ∗(t) in the dual vector space satisfying the equations Γ · Γ∗ = 0, Γ0 · Γ∗ = 0, . . . , Γ(n−1) · Γ∗ = 0, (1.1.2) where dot denotes the pairing between vectors and covectors. Any solution Γ∗(t) of (1.1.2) projects to γ∗(t). Since γ is non-degenerate, the rank of system (1.1.2) equals n. Therefore, γ∗(t) is uniquely defined and depends smoothly on t. Differentiating system (1.1.2), we see that Γ(i) ·Γ∗(j) = 0 for i+j ≤ n−1. Hence the osculating flag of the curve γ∗ is dual to that of γ and the curve γ∗ is non-degenerate. In particular, for i = 0, we obtain Γ · Γ∗(j) = 0 with j = 0, . . . , n − 1. Thus, (γ∗)∗ = γ. As in the 2-dimensional case, the dual curve γ∗ can be also obtained as the envelope of a 1-parameter family of subspaces in RPn∗ , namely, of the dual k-th osculating spaces of γ. All this is illustrated by the following celebrated example. Example 1.1.4. Consider a curve γ(t) in RP3 given, in affine coordinates, by the equations: y1 = t, y2 = t2 , y3 = t4 . This curve is non-degenerate at point γ(0). The plane, dual to point γ(t), is given, in an appropriate affine coordinate system (a1, a2, a3) in RP3∗ , by
- 19. 1.2. DISCRETE INVARIANTS AND CONFIGURATIONS 5 the equation t4 + a1t2 + a2t + a3 = 0. (1.1.3) This 1-parameter family of planes envelops a surface called the swallow tail and shown in figure 1.4. This developable surface consists of the tangent lines to the curve γ∗. Note the cusp of γ∗ at the origin. Figure 1.4: Swallow tail Comment The study of polynomials (1.1.3) and figure 1.4 go back to the XIX-th cen- tury [118]; the name “swallow tail” was invented by R. Thom in mid XX-th century in the framework of the emerging singularity theory (see [16]). The swallow tail is the set of polynomials (1.1.3) with multiple roots, and the curve γ∗ corresponds to polynomials with triple roots. This surface is a typical example of a developable surface, i.e., surface of zero Gauss curva- ture. The classification of developable surfaces is due to L. Euler (cf.[193]): generically, such a surface consists of the tangent lines of a curve, called the edge of regression. The edge of regression itself has a singularity as in figure 1.4. Unlike the Plücker formula of classic algebraic geometry, the result of Exercise 1.1.2 is surprisingly recent; it was obtained by Fabricius-Bjerre in 1962 [61]. This result has numerous generalizations, see, e.g., [199, 66]. 1.2 Discrete invariants and configurations The oldest invariants in projective geometry are projective invariants of configurations of point and lines. Our exposition is just a brief excursion to the subject, for a thorough treatment see, e.g., [22].
- 20. 6 CHAPTER 1. INTRODUCTION Cross-ratio Consider the projective line RP1 . Every triple of points can be taken to any other triple by a projective transformation. This is not the case for quadruples of points: four points in RP1 have a numeric invariant called the cross-ratio. Choosing an affine parameter t to identify RP1 with R ∪ {∞}, the action of PGL(2, R) is given by fractional-linear transformations: t 7→ at + b ct + d . (1.2.1) The four points are represented by numbers t1, t2, t3, t4, and the cross-ratio is defined as [t1, t2, t3, t4] = (t1 − t3)(t2 − t4) (t1 − t2)(t3 − t4) . (1.2.2) A quadruple of points is called harmonic if its cross-ratio is equal to −1. Exercise 1.2.1. a) Check that the cross-ratio does not change under trans- formations (1.2.1). b) Investigate how the cross-ratio changes under permutations of the four points. A B C D A B C D a b c d ' ' ' ' Figure 1.5: Cross-ratio of lines: [A, B, C, D] = [A0, B0, C0, D0] := [a, b, c, d] One defines also the cross-ratio of four concurrent lines in RP2 , that is, four lines through one point. The pencil of lines through a point identifies with RP1 , four lines define a quadruple of points in RP1 , and we take their cross-ratio. Equivalently, intersect the four lines with an auxiliary line and take the cross-ratio of the intersection points therein, see figure 1.5.
- 21. 1.2. DISCRETE INVARIANTS AND CONFIGURATIONS 7 Pappus and Desargues Let us mention two configurations in the projective plane. Figures 1.6 depict two classical theorems. Figure 1.6: Pappus and Desargues theorems The Pappus theorem describes the following construction which we rec- ommend to the reader to perform using a ruler or his favorite drawing soft- ware. Start with two lines, pick three points on each. Connect the points pairwise as shown in figure 1.6 to obtain three new intersection points. These three points are also collinear. In the Desargues theorem, draw three lines through one point and pick two points on each to obtain two perspective triangles. Intersect the pairs of corresponding sides of the triangles. The three points of intersection are again collinear. Pascal and Brianchon The next theorems, depicted in figure 1.7, involve conics. To obtain the Pascal theorem, replace the two original lines in the Pappus configuration by a conic. In the Brianchon theorem, circumscribe a hexagon about a conic and connect the opposite vertices by diagonals. The three lines intersect at one point. Unlike the Pappus and Desargues configurations, the Pascal and Brian- chon ones are projectively dual to each other. Steiner Steiner’s theorem provides a definition of the cross-ratio of four points on a conic. Choose a point P on a conic. Given four points A, B, C, D, define their cross-ratio as that of the lines (PA), (PB), (PC), (PD). The theorem
- 22. 8 CHAPTER 1. INTRODUCTION Figure 1.7: Pascal and Brianchon theorems A D C B P P1 Figure 1.8: Steiner theorem asserts that this cross-ratio is independent of the choice of point P: [(PA), (PB), (PC), (PD)] = [(P1A), (P1B), (P1C), (P1D)] in figure 1.8. Comment In 1636 Girard Desargues published a pamphlet “A sample of one of the general methods of using perspective” that laid the foundation of projective geometry; the Desargues theorem appeared therein. The Pappus configura- tion is considerably older; it was known as early as the III-rd century A.D. The triple of lines in figure 1.6 is a particular case of a cubic curve, the Pappus configuration holds true for 9 points on an arbitrary cubic curve – see figure 1.9. This more general formulation contains the Pascal theorem as well. Particular cases of Steiner’s theorem were already known to Apol-
- 23. 1.3. INTRODUCING SCHWARZIAN DERIVATIVE 9 Figure 1.9: Generalized Pappus theorem lonius 1. Surprisingly, even today, there appear new generalizations of the Pappus and the Desargues theorems, see [182, 183]. 1.3 Introducing Schwarzian derivative Projective differential geometry studies projective invariants of functions, diffeomorphisms, submanifolds, etc. One way to construct such invariants is to investigate how discrete invariants vary in continuous families. Schwarzian derivative and cross-ratio The best known and most popular projective differential invariant is the Schwarzian derivative. Consider a diffeomorphism f : RP1 → RP1 . The Schwarzian derivative measures how f changes the cross-ratio of infinitesi- mally close points. Let x be a point in RP1 and v be a tangent vector to RP1 at x. Extend v to a vector field in a vicinity of x and denote by φt the corresponding local one-parameter group of diffeomorphisms. Consider 4 points: x, x1 = φε(x), x2 = φ2ε(x), x3 = φ3ε(x) 1 We are indebted to B. A. Rosenfeld for enlightening discussions on Ancient Greek mathematics
- 24. 10 CHAPTER 1. INTRODUCTION (ε is small) and compare their cross-ratio with that of their images under f. It turns out that the cross-ratio does not change in the first order in ε: [f(x), f(x1), f(x2), f(x3)] = [x, x1, x2, x3] − 2ε2 S(f)(x) + O(ε3 ). (1.3.1) The ε2–coefficient depends on the diffeomorphism f, the point x and the tangent vector v, but not on its extension to a vector field. The term S(f) is called the Schwarzian derivative of a diffeomorphism f. It is homogeneous of degree 2 in v and therefore S(f) is a quadratic differential on RP1 , that is, a quadratic form on TRP1 . Choose an affine coordinate x ∈ R ∪ {∞} = RP1 . Then the projective transformations are identified with fractional-linear functions and quadratic differentials are written as φ = a(x) (dx)2. The change of variables is then described by the formula φ ◦ f = f0 2 a(f(x)) (dx)2 . (1.3.2) The Schwarzian derivative is given by the formula S(f) = f000 f0 − 3 2 f00 f0 2 ! (dx)2 . (1.3.3) Exercise 1.3.1. a) Check that (1.3.1) contains no term, linear in ε. b) Prove that S(f) does not depend on the extension of v to a vector field. c) Verify formula (1.3.3). The Schwarzian derivative enjoys remarkable properties. • By the very construction, S(g) = 0 if g is a projective transformation, and S(g ◦ f) = S(f) if g is a projective transformation. Conversely, if S(g) = 0 then g is a projective transformation. • For arbitrary diffeomorphisms f and g, S(g ◦ f) = S(g) ◦ f + S(f) (1.3.4) where S(g) ◦ f is defined as in (1.3.2). Homological meaning of this equation will be explained in Section 1.5. Exercise 1.3.2. Prove formula (1.3.4).
- 25. 1.3. INTRODUCING SCHWARZIAN DERIVATIVE 11 Curves in the projective line By a curve we mean a parameterized curve, that is, a smooth map from R to RP1 . In other words, we consider a moving one-dimensional subspace in R2. Two curves γ1(t) and γ2(t) are called equivalent if there exists a projective transformation g ∈ PGL(2, R) such that γ2(t) = g◦γ1(t). Recall furthermore that a curve in RP1 is non-degenerate if its speed is never vanishing (cf. Section 1.1). One wants to describe the equivalence classes of non-degenerate curves in RP1 . In answering this question we encounter, for the first time, a powerful tool of projective differential geometry, linear differential operators. Theorem-construction 1.3.1. There is a one-to-one correspondence be- tween equivalence classes of non-degenerate curves in RP1 and Sturm-Liou- ville operators L = d2 dt2 + u(t) (1.3.5) where u(t) is a smooth function. Proof. Consider the Sturm-Liouville equation ψ̈(t)+u(t)ψ(t) = 0 associated with an operator (1.3.5). The space of solutions, V , of this equation is two- dimensional. Associating to each value of t a one-dimensional subspace of V consisting of solutions vanishing for this t, we obtain a family of one- dimensional subspaces depending on t. Finally, identifying V with R2 by an arbitrary choice of a basis, ψ1(t), ψ2(t), we obtain a curve in RP1 , defined up to a projective equivalence. Γ(t) γ(t) 0 Γ(t) Figure 1.10: Canonical lift of γ to R2: the area |Γ(t), Γ̇(t)| = 1 Conversely, consider a non-degenerate curve γ(t) in RP1 . It can be uniquely lifted to R2 as a curve Γ(t) such that |Γ(t), Γ̇(t)| = 1, see figure 1.10. Differentiate to see that the vector Γ̈(t) is proportional to Γ(t): Γ̈(t) + u(t)Γ(t) = 0.
- 26. 12 CHAPTER 1. INTRODUCTION We have obtained a Sturm-Liouville operator. If γ(t) is replaced by a pro- jectively equivalent curve then its lift Γ(t) is replaced by a curve A(Γ(t)) where A ∈ SL(2, R), and the respective Sturm-Liouville operator remains intact. Exercise 1.3.3. a) The curve corresponding to a Sturm-Liouville operator is non-degenerate. b) The two above constructions are inverse to each other. To compute explicitly the correspondence between Sturm-Liouville op- erators and non-degenerate curves, fix an affine coordinate on RP1 . A curve γ is then given by a function f(t). Exercise 1.3.4. Check that u(t) = 1 2 S(f(t)). Thus the Schwarzian derivative enters the plot for the second time. Projective structures on R and S1 The definition of projective structure resembles many familiar definitions in differential topology or differential geometry (smooth manifold, vector bundle, etc.). A projective structure on R is given by an atlas (Ui, ϕi) where (Ui) is an open covering of R and the maps ϕi : Ui → RP1 are local diffeomorphisms satisfying the following condition: the locally defined maps ϕi ◦ϕ−1 j on RP1 are projective. Two such atlases are equivalent if their union is again an atlas. Informally speaking, a projective structure is a local identification of R with RP1 . For every quadruple of sufficiently close points one has the notion of cross-ratio. A projective atlas defines an immersion ϕ : R → RP1 ; a projective struc- ture gives a projective equivalence class of such immersions. The immersion ϕ, modulo projective equivalence, is called the developing map . According to Theorem 1.3.1, the developing map ϕ gives rise to a Sturm-Liouville oper- ator (1.3.5). Therefore, the space of projective structures on S1 is identified with the space of Sturm-Liouville operators. The definition of projective structure on S1 is analogous, but it has a new feature. Identifying S1 with R/Z, the developing map satisfies the following condition: ϕ(t + 1) = M(ϕ(t)) (1.3.6) for some M ∈ PGL(2, R). The projective map M is called the monodromy . Again, the developing map is defined up to the projective equivalence: (ϕ(t), M) ∼ (gϕ(t), gMg−1) for g ∈ PGL(2, R).
- 27. 1.3. INTRODUCING SCHWARZIAN DERIVATIVE 13 The monodromy condition (1.3.6) implies that, for the corresponding Sturm-Liouville operator, one has u(t + 1) = u(t), while the solutions have the monodromy f M ∈ SL(2, R), which is a lift of M. To summarize, the space of projective structures on S1 is identified with the space of Sturm-Liouville operators with 1-periodic potentials u(t). Diff(S1 )- and Vect(S1 )-action on projective structures The group of diffeomorphisms Diff(S1) naturally acts on projective atlases and, therefore, on the space of projective structures. In terms of the Sturm- Liouville operators, this action is given by the transformation rule for the potential Tf−1 : u 7→ f0 2 u(f) + 1 2 S(f), (1.3.7) where f ∈ Diff(S1). This follows from Exercise 1.3.4 and formula (1.3.4). The Lie algebra corresponding to Diff(S1) is the algebra of vector fields Vect(S1). The vector fields are written as X = h(t)d/dt and their commu- tator as [X1, X2] = h1h0 2 − h0 1h2 d dt . Whenever one has a differentiable action of Diff(S1), one also has an action of Vect(S1) on the same space. Exercise 1.3.5. Check that the action of a vector field X = h(t)d/dt on the potential of a Sturm-Liouville operator is given by tX : u 7→ hu0 + 2h0 u + 1 2 h000 . (1.3.8) It is interesting to describe the kernel of this action. Exercise 1.3.6. a) Let φ1 and φ2 be two solutions of the Sturm-Liouville equation φ00(t)+u(t)φ(t) = 0. Check that, for the vector field X = φ1φ2 d/dt, one has tX = 0. b) The kernel of the action t is a Lie algebra isomorphic to sl(2, R); this is precisely the Lie algebra of symmetries of the projective structure corre- sponding to the Sturm-Liouville operator. Hint. The space of solutions of the equation tX = 0 is three-dimensional, hence the products of two solutions of the Sturm-Liouville equation span this space.
- 28. 14 CHAPTER 1. INTRODUCTION Γ y1 y2 y1 y2 Figure 1.11: Zeroes of solutions Sturm theorem on zeroes The classic Sturm theorem states that between two zeroes of a solution of a Sturm-Liouville equation any other solution has a zero as well. The simplest proof is an application of the above identification between Sturm-Liouville equations and projective structures on S1. Consider the corresponding de- veloping map γ : S1 → RP1 and its lift Γ to R2. Every solution φ of the Sturm-Liouville equation is a pull-back of a linear function y on R2. Ze- roes of φ are the intersection points of Γ with the line y = 0. Since γ is non-degenerate, the intermediate value theorem implies that between two intersections of Γ with any line there is an intersection with any other line, see figure 1.11 and [163] for an elementary exposition. Comment The Schwarzian derivative is historically the first and most fundamental projective differential invariant. The natural identification of the space of projective structures with the space of Sturm-Liouville operators is an im- portant conceptual result of one-dimensional projective differential geome- try, see [222] for a survey. Exercise 1.3.6 is Kirillov’s observation [115]. 1.4 Further example of differential invariants: pro- jective curvature The second oldest differential invariant of projective geometry is the pro- jective curvature of a plane curve. The term “curvature” is somewhat mis- leading: the projective curvature is, by no means, a function on the curve. We will define the projective curvature as a projective structure on the curve. In a nutshell, the curve is approximated by its osculating conic which, by
- 29. 1.4. FURTHER EXAMPLE OF DIFFERENTIAL INVARIANTS: PROJECTIVE CURVATURE15 Steiner’s theorem (cf. Section 1.2), has a projective structure induced from RP2 ; this projective structure is transplanted from the osculating conic to the curve. To realize this program, we will proceed in a traditional way and represent projective curves by differential operators. Plane curves and differential operators Consider a parameterized non-degenerate curve γ(t) in RP2 , that is, a curve without inflection points (see Section 1.1 for a general definition). Repeating the construction of Theorem 1.3.1 yields a third-order linear differential operator A = d3 dt3 + q(t) d dt + r(t). (1.4.1) Example 1.4.1. Let γ(t) be the conic (recall that all non-degenerate conics in RP2 are projectively equivalent). The corresponding differential operator (1.4.1) has a special form: A1 = d3 dt3 + q(t) d dt + 1 2 q0 (t). (1.4.2) Indeed, consider the Veronese map V : RP1 → RP2 given by the formula V (x0 : x1) = (x2 0 : x0x1 : x2 1). (1.4.3) The image of RP1 is a conic, and γ(t) is the image of a parameterized curve in RP1 . A parameterized curve in RP1 corresponds to a Sturm-Liouville operator (1.3.5) so that {x0(t), x1(t)} is a basis of solutions of the Sturm- Liouville equation Lψ = 0. It remains to check that every product y(t) = xi(t)xj(t), i, j = 1, 2 satisfies A1y = 0 with q(t) = 4u(t). Exercise 1.4.2. We now have two projective structures on the conic in RP2 : the one given by Steiner’s theorem and the one induced by the Veronese map from RP1 . Prove that these structures coincide. Projective curvature via differential operators Associate the following Sturm-Liouville operator with the operator A: L = d2 dt2 + 1 4 q(t). (1.4.4)
- 30. 16 CHAPTER 1. INTRODUCTION According to Section 1.3, we obtain a projective structure on R and thus on the parameterized curve γ(t). Theorem 1.4.3. This projective structure on γ(t) does not depend on the choice of the parameter t. Proof. Recall the notion of dual (or adjoint) operator: for a differential monomial one has a(t) dk dtk ∗ = (−1)k dk dtk ◦ a(t). (1.4.5) Consider the decomposition of the operator (1.4.1) into the sum A = A1 + A0 (1.4.6) of its skew-symmetric part A1 = −A∗ 1 given by (1.4.2) and the symmetric part A0 = A∗ 0. Note, that the symmetric part is a scalar operator: A0 = r(t) − 1 2 q0 (t). (1.4.7) The decomposition (1.4.6) is intrinsic, that is, independent of the choice of the parameter t (cf. Section 2.2 below). The correspondence A 7→ L is a composition of two operations: A 7→ A1 and A1 7→ L; the second one is also intrinsic, cf. Example 1.4.1. Exercise 1.4.4. The operator (1.4.2) is skew-symmetric: A∗ 1 = −A1. To wit, a non-degenerate curve in RP2 carries a canonical projective structure which we call the projective curvature. In the next chapter we will explain that the expression A0 in (1.4.7) is, in fact, a cubic differential; the cube root (A0)1/3 is called the projective length element . The projective length element is identically zero for a conic and, moreover, vanishes in those points of the curve in which the osculating conic is hyper-osculating. Traditionally, the projective curvature is considered as a function q(t) where t is a special parameter for which A0 ≡ 1, i.e., the projective length element equals dt. On the other hand, one can choose a different parameter x on the curve in such a way that q(x) ≡ 0, namely, the affine coordinate of the defined projective structure. This shows that the projective curvature is neither a function nor a tensor.
- 31. 1.4. FURTHER EXAMPLE OF DIFFERENTIAL INVARIANTS: PROJECTIVE CURVATURE17 Exercise 1.4.5. a) Let A be the differential operator corresponding to a non-degenerate parameterized curve γ(t) in RP2 . Prove that the operator corresponding to the dual curve γ∗(t) is −A∗. b) Consider a non-degenerate parameterized curve γ(t) in RP2 and let γ∗(t) be projectively equivalent to γ(t), i.e., there exists a projective isomorphism ϕ : RP2 → RP2∗ such that γ∗(t) = ϕ(γ(t)). Prove that γ(t) is a conic. l l l l 0 1 2 3 Figure 1.12: Projective curvature as cross-ratio Projective curvature and cross-ratio Consider four points γ(t), γ(t + ε), γ(t + 2ε), γ(t + 3ε) of a non-degenerate curve in RP2 . These points determine four lines `0, `1, `2 and `3 as in figure 1.12. Let us expand the cross-ratio of these lines in powers of ε. Exercise 1.4.6. One has [`0, `1, `2, `3] = 4 − 2ε2 q(t) + O(ε3 ). (1.4.8) This formula relates the projective curvature with the cross-ratio. Comparison with affine curvature Let us illustrate the preceding construction by comparison with geometri- cally more transparent notion of the affine curvature and the affine param- eter (see, e.g., [193]).
- 32. 18 CHAPTER 1. INTRODUCTION x x+ε v Figure 1.13: Cubic form on an affine curve Consider a non-degenerate curve γ in the affine plane with a fixed area form. We define a cubic form on γ as follows. Let v be a tangent vector to γ at point x. Extend v to a tangent vector field along γ and denote by φt the corresponding local one-parameter group of diffeomorphisms of γ. Consider the segment between x and φε(x), see figure 1.13, and denote by A(x, v, ε) the area, bounded by it and the curve. This function behaves cubically in ε, and we define a cubic form σ(x, v) = lim ε→0 A(x, v, ε) ε3 . (1.4.9) A parameter t on γ is called affine if σ = c(dt)3 with a positive constant c. By the very construction, the notion of affine parameter is invariant with respect to the group of affine transformations of the plane while σ is invariant under the (smaller) equiaffine group. Alternatively, an affine parameter is characterized by the condition |γ0 (t), γ00 (t)| = const. Hence the vectors γ000(t) and γ0(t) are proportional: γ000(t) = −k(t)γ0(t). The function k(t) is called the affine curvature. The affine parameter is not defined at inflection points. The affine cur- vature is constant if and only if γ is a conic. Comment The notion of projective curvature appeared in the literature in the second half of the XIX-th century. From the very beginning, curves were studied in the framework of differential operators – see [231] for an account of this early period of projective differential geometry. In his book [37], E. Cartan also calculated the projective curvature as a function of the projective length parameter. However, he gave an inter- pretation of the projective curvature in terms of a projective structure on the curve. Cartan invented a geometrical construction of developing a non- degenerate curve on its osculating conic. This construction is a projective
- 33. 1.5. SCHWARZIAN DERIVATIVE AS A COCYCLE OF DIFF(RP1 ) 19 counterpart of the Huygens construction of the involute of a plane curve using a non-stretchable string: the role of the tangent line is played by the osculating conic and the role of the Euclidean length by the projective one. Affine differential geometry and the corresponding differential invariants appeared later than the projective ones. A systematic theory was developed between 1910 and 1930, mostly by Blaschke’s school. 1.5 Schwarzian derivative as a cocycle of Diff(RP1 ) The oldest differential invariant of projective geometry, the Schwarzian deri- vative, remains the most interesting one. In this section we switch gears and discuss the relation of the Schwarzian derivative with cohomology of the group Diff(RP1 ). This contemporary viewpoint leads to promising ap- plications that will be discussed later in the book. To better understand the material of this and the next section, the reader is recommended to consult Section 8.4. Invariant and relative 1-cocycles Let G be a group, V a G-module and T : G → End(V ) the G-action on V . A map C : G → V is called a 1-cocycle on G with coefficients in V if it satisfies the condition C(gh) = Tg C(h) + C(g). (1.5.1) A 1-cocycle C is called a coboundary if C(g) = Tg v − v (1.5.2) for some fixed v ∈ V . The quotient group of 1-cocycles by coboundaries is H1(G, V ), the first cohomology group; see Section 8.4 for more details. Let H be a subgroup of G. A 1-cocycle C is H-invariant if C(hgh−1 ) = Th C(g) (1.5.3) for all h ∈ H and g ∈ G. Another important class of 1-cocycles associated with a subgroup H consists of the cocycles vanishing on H. Such cocycles are called H-relative . Exercise 1.5.1. Let H be a subgroup of G and let C be a 1-cocycle on G. Prove that the following three conditions are equivalent:
- 34. 20 CHAPTER 1. INTRODUCTION 1) C(h) = 0 for all h ∈ H; 2) C(gh) = C(g) for all h ∈ H and g ∈ G; 3) C(hg) = Th (C(g)) for all h ∈ H and g ∈ G. The property of a 1-cocycle to be H-relative is stronger than the condition to be H-invariant. Exercise 1.5.2. Check that the conditions 1) – 3) imply (1.5.3). Tensor densities in dimension 1 All tensor fields on a one-dimensional manifold M are of the form: φ = φ(x)(dx)λ , (1.5.4) where λ ∈ R and x is a local coordinate; φ is called a tensor density of degree λ. The space of tensor densities is denoted by Fλ(M), or Fλ, for short. Equivalently, a tensor density of degree λ is defined as a section of the line bundle (T ∗M)⊗λ. The group Diff(M) naturally acts on Fλ. To describe explicitly this action, consider the space of functions C∞(M) and define a 1-parameter family of Diff(M)-actions on this space: Tλ f−1 : φ(x) 7→ f0 λ φ(f(x)) , f ∈ Diff(M) (1.5.5) cf. formula (1.3.2) for quadratic differentials. The Diff(M)-module Fλ is nothing else but the module (C∞(M), Tλ). Although all Fλ are isomorphic to each other as vector spaces, Fλ and Fµ are not isomorphic as Diff(M)- modules unless λ = µ (cf. [72]). In the case M = S1, there is a Diff(M)-invariant pairing Fλ ⊗F1−λ → R given by the integral hφ(x)(dx)λ , ψ(x)(dx)1−λ i = Z S1 φ(x)ψ(x)dx Example 1.5.3. In particular, F0 is the space of smooth functions, F1 is the space of 1-forms, F2 is the space of quadratic differentials, familiar from the definition of the Schwarzian derivative, while F−1 is the space of vector fields. The whole family Fλ is of importance, especially for integer and half-integer values of λ. Exercise 1.5.4. a) Check that formula (1.5.5) indeed defines an action of Diff(M), that is, for all diffeomorphisms f, g, one has Tλ f ◦ Tλ g = Tλ f◦g. b) Show that the Vect(M)-action on Fλ(M) is given by the formula Lλ h(x) d dx : φ(dx)λ 7→ (hφ0 + λ h0 φ)(dx)λ . (1.5.6)
- 35. 1.5. SCHWARZIAN DERIVATIVE AS A COCYCLE OF DIFF(RP1 ) 21 First cohomology with coefficients in tensor densities Recall identity (1.3.4) for the Schwarzian derivative. This identity means that the Schwarzian derivative defines a 1-cocycle f 7→ S(f −1) on Diff(RP1 ) with coefficients in the space of quadratic differentials F2(RP1 ). This cocycle is not a coboundary; indeed, unlike S(f), any coboundary (1.5.2) depends only on the 1-jet of a diffeomorphism – see formula (1.5.5). The Schwarzian derivative vanishes on the subgroup PGL(2, R), and thus it is PGL(2, R)-invariant. Let us describe the first cohomology of the group Diff(RP1 ) with coeffi- cients in Fλ. These cohomologies can be interpreted as equivalence classes of affine modules (or extensions) on Fλ. If G is a Lie group and V its module then a structure of affine module on V is a structure of G-module on the space V ⊕ R defined by e Tg : (v, α) 7→ (Tg v + α C(g), α), where C is a 1-cocycle on G with values in V . See Section 8.4 for more information on affine modules and extensions. Theorem 1.5.5. One has H1 (Diff(RP1 ); Fλ) = ( R, λ = 0, 1, 2, 0, otherwise (1.5.7) We refer to [72] for details. The corresponding cohomology classes are represented by the 1-cocycles C0(f−1 ) = ln f0 , C1(f−1 ) = f00 f0 dx, C2(f−1 ) = f000 f0 − 3 2 f00 f0 2 (dx)2 . The first cocycle makes sense in Euclidean geometry and the second one in affine geometry. Their restrictions to the subgroup PGL(2, R) are non- trivial, hence these cohomology classes cannot be represented by PGL(2, R)- relative cocycles. One is usually interested in cohomology classes, not in the represent- ing cocycles which, as a rule, depend on arbitrary choices. However, the Schwarzian derivative is canonical in the following sense. Theorem 1.5.6. The Schwarzian derivative is a unique (up to a constant) PGL(2, R)-relative 1-cocycle on Diff(RP1 ) with coefficients in F2.
- 36. 22 CHAPTER 1. INTRODUCTION Proof. If there are two such cocycles, then, by Theorem 1.5.5, their linear combination is a coboundary, and this coboundary vanishes on PGL(2, R). Every coboundary is of the form C(f) = Tf (φ) − φ for some φ ∈ F2. Therefore one has a non-zero PGL(2, R)-invariant quadratic differential. It remains to note that PGL(2, R) does not preserve any tensor field on RP1 . Exercise 1.5.7. Prove that the infinitesimal version of the Schwarzian derivative is the following 1-cocycle on the Lie algebra Vect(RP1 ): h(x) d dx 7→ h000 (x) (dx)2 . (1.5.8) 1.6 Virasoro algebra: the coadjoint representation The Virasoro algebra is one of the best known infinite-dimensional Lie al- gebras, defined as a central extension of Vect(S1). A central extension of a Lie algebra g is a Lie algebra structure on the space g ⊕ R given by the commutator [(X, α), (Y, β)] = ([X, Y ], c(X, Y )), where X, Y ∈ g, α, β ∈ R and c : g → R is a 1-cocycle. The reader can find more information on central extensions in Section 8.4. Definition of the Virasoro algebra The Lie algebra Vect(S1) has a central extension given by the so-called Gelfand-Fuchs cocycle c h1(x) d dx , h2(x) d dx = Z S1 h0 1(x) h00 2(x) dx. (1.6.1) The corresponding Lie algebra is called the Virasoro algebra and will be denoted by Vir. This is a unique (up to isomorphism) non-trivial central extension of Vect(S1) (cf. Lemma 8.5.3). Exercise 1.6.1. Check the Jacobi identity for Vir. Note that the cocycle (1.6.1) is obtained by pairing the cocycle (1.5.8) with a vector field.
- 37. 1.6. VIRASORO ALGEBRA: THE COADJOINT REPRESENTATION23 Computing the coadjoint representation To explain the relation of the Virasoro algebra to projective geometry we use the notion of coadjoint representation defined as follows. A Lie algebra g acts on its dual space by had∗ X φ, Y i := −hφ, [X, Y ]i, for φ ∈ g∗ and X, Y ∈ g. This coadjoint representation carries much infor- mation about the Lie algebra. The dual space to the Virasoro algebra is Vir∗ = Vect(S1)∗ ⊕ R. It is always natural to begin the study of the dual space to a functional space with its subspace called the regular dual. This subspace is spanned by the distributions given by smooth compactly supported functions. Consider the regular dual space, Vir∗ reg = C∞(S1)⊕R consisting of pairs (u(x), c) where u(x) ∈ C∞(S1) and c ∈ R, so that h(u(x), c), (h(x)d/dx, α)i := Z S1 u(x)h(x)dx + cα. The regular dual space is invariant under the coadjoint action. Exercise 1.6.2. The explicit formula for the coadjoint action of the Vira- soro algebra on its regular dual space is ad∗ (hd/dx, α)(u, c) = (hu0 + 2h0 u − c h000 , 0). (1.6.2) Note that the center of Vir acts trivially. A remarkable coincidence In the first two terms of the above formula (1.6.2) we recognize the Lie derivative (1.5.6) of quadratic differentials, the third term is nothing else but the cocycle (1.5.8), so that the action (1.6.2) is an affine module (see Section 1.5). Moreover, this action coincides with the natural Vect(S1)- action on the space of Sturm-Liouville operators (for c = −1/2), see formula (1.3.8). Thus one identifies, as Vect(S1)-modules, the regular dual space Vir∗ reg and the space of Sturm-Liouville operators (u(x), c) ↔ −2c d2 dx2 + u(x) (1.6.3) and obtains a nice geometrical interpretation for the coadjoint representation of the Virasoro algebra.
- 38. 24 CHAPTER 1. INTRODUCTION Remark 1.6.3. To simplify exposition, we omit the definition of the Vira- soro group (the group analog of the Virasoro algebra) and the computation of its coadjoint action which, indeed, coincides with the Diff(S1)-action (1.3.7). Coadjoint orbits The celebrated Kirillov’s orbit method concerns the study of the coadjoint representation. Coadjoint orbits of a Lie algebra g are defined as integral surfaces in g∗, tangent to the vector fields φ̇ = ad∗ Xφ for all X ∈ g 2. Classification of the coadjoint orbits of a Lie group or a Lie algebra is always an interesting problem. The identification (1.6.3) makes it possible to express invariants of the coadjoint orbits of the Virasoro algebra in terms of invariants of Sturm- Liouville operators (and projective structures on S1, see Theorem 1.3.1). An invariant of a differential operator on S1 is the monodromy operator mentioned in Section 1.3. In the case of Sturm-Liouville operators, this is an element of the universal covering ^ PGL(2, R). Theorem 1.6.4. The monodromy operator is the unique invariant of the coadjoint orbits of the Virasoro algebra. Proof. Two elements (u0(x), c) and (u1(x), c) of Vir∗ reg belong to the same coadjoint orbit if and only if there is a one-parameter family (ut(x), c) with t ∈ [0, 1] such that, for every t, the element ( ˙ ut(x), 0) is the result of the coadjoint action of Vir; here dot denotes the derivative with respect to t. In other words, there exists ht(x) d dx ∈ Vect(S1) such that u̇t(x) = ht(x) u0 t(x) + 2h0 t(x) ut(x) − c h000 t (x). (1.6.4) According to (1.6.3), a family (ut(x), c) defines a family of Sturm-Liouville operators: Lt = −2c(d/dx)2 + u(x)t. Consider the corresponding family of Sturm-Liouville equations Lt(φ) = −2c φ00 (x) + u(x)t φ(x) = 0. For every t, one has a two-dimensional space of solutions, hφ1t(x), φ2t(x)i. Define a Vect(S1)-action on the space of solutions using the Leibnitz rule: (ad∗ h d dx L)(φ) + L(Th d dx φ) = 0 2 This definition allows us to avoid using the notion of a Lie group, and sometimes this simplifies the situation, for instance, in the infinite-dimensional case.
- 39. 1.6. VIRASORO ALGEBRA: THE COADJOINT REPRESENTATION25 where the Vect(S1)-action on the space of Sturm-Liouville operators is given by formula (1.6.2). It turns out, that the solutions of Sturm-Liouville equa- tions behave as tensor densities of degree −1 2 . Exercise 1.6.5. Check that, in the above formula, Th d dx = L − 1 2 h d dx , where Lλ h d dx is the Lie derivative of a λ-density defined by (1.5.6). To solve the (nonlinear) “homotopy” equation (1.6.4), it suffices now to find a family of vector fields ht(x) d dx such that L − 1 2 ht d dx φ1t = ht φ1 0 t − 1 2 h0 t φ1t = ˙ φ1t L − 1 2 ht d dx φ2t = ht φ2 0 t − 1 2 h0 t φ2t = ˙ φ2t This is just a system of linear equations in two variables, ht(x) and h0 t(x), with the solution ht(x) = ˙ φ1t ˙ φ2t φ1t φ2t h0 t(x) = 2 ˙ φ1t ˙ φ2t φ1 0 t φ2 0 t . (1.6.5) One can choose a basis of solutions hφ1t(x), φ2t(x)i so that the Wronski determinant is independent of t: φ1t φ2t φ1 0 t φ2 0 t ≡ 1. Then one has ˙ φ1t ˙ φ2t φ1 0 t φ2 0 t = ˙ φ1t 0 ˙ φ2t 0 φ1t φ2t It follows that the first formula in (1.6.5) implies the second. Finally, if the monodromy operator of a family of Sturm-Liouville oper- ators Lt does not depend on t, then one can choose a basis hφ1t(x), φ2t(x)i in such a way that the monodromy matrix, say M, in this basis does not depend on t. Then one concludes from (1.6.5) that ht(x + 2π) = det M · ht(x) = ht(x), since M ∈ ^ SL(2, R). Therefore, ht(x)d/dx is, indeed, a family of vector fields on S1.
- 40. 26 CHAPTER 1. INTRODUCTION Remark 1.6.6. One can understand, in a more traditional way, the mon- odromy operator as an element of SL(2, R), instead of its universal covering. Then there is another discrete invariant, representing a class in π1(SL(2, R)). This invariant is nothing but the winding number of the corresponding curve in RP1 , see Section 1.3. For instance, there are infinitely many connected components in the space of Sturm-Liouville operators with the same mon- odromy. Relation to infinite-dimensional symplectic geometry A fundamental fact which makes the notion of coadjoint orbits so impor- tant (in comparison with the adjoint orbits) is that every coadjoint orbit has a canonical g-invariant symplectic structure (often called the Kirillov symplectic form). Moreover, the space g∗ has a Poisson structure called the Lie-Poisson(-Berezin-Kirillov-Kostant-Souriau) bracket, and the coad- joint orbits are the corresponding symplectic leaves. See Section 8.2 for a brief introduction to symplectic and Poisson geometry. An immediate corollary of the above remarkable coincidence is that the space of the Sturm-Liouville operators is endowed with a natural Diff(S1)- invariant Poisson structure; furthermore, it follows from Theorem 1.6.4 that the space of Sturm-Liouville operators with a fixed monodromy is an (infinite dimensional) symplectic manifold. Comment The Virasoro algebra was discovered in 1967 by I. M. Gelfand and D. B. Fuchs. It appeared in the physical literature around 1975 and became very popular in conformal field theory (see [90] for a comprehensive reference). The coadjoint representation of Lie groups and Lie algebras plays a spe- cial role in symplectic geometry and representation theory, cf. [112]. The observation relating the coadjoint representation to the natural Vect(S1)- action on the space of Sturm-Liouville operators and, therefore, on the space of projective structures on S1, and the classification of the coadjoint orbits was made in 1980 independently by A. A. Kirillov and G. Segal [116, 186]. The classification of the coadjoint orbits then follows from the classical work by Kuiper [126] (see also [130]) on classification of projective structures. Our proof, using the homotopy method, is probably new. This and other remarkable properties of the Virasoro algebra, its relation with the Korteweg-de Vries equation, moduli spaces of holomorphic curves, etc., make this infinite-dimensional Lie algebra one of the most interesting
- 41. 1.6. VIRASORO ALGEBRA: THE COADJOINT REPRESENTATION27 objects of modern mathematics and mathematical physics.
- 42. 28 CHAPTER 1. INTRODUCTION
- 43. Chapter 2 Geometry of projective line What are geometric objects? On the one hand, curves, surfaces, various geometric structures; on the other, tensor fields, differential operators, Lie group actions. The former objects originated in classical geometry while the latter ones are associated with algebra. Both points of view are legitimate, yet often separated. This chapter illustrates unity of geometric and algebraic approaches. We study geometry of a simple object, the projective line. Such notions as non- degenerate immersions of a line in projective space and linear differential operators on the line are intrinsically related, and this gives two comple- mentary viewpoints on the same thing. Following F. Klein, we understand geometry in terms of group actions. In the case of the projective line, two groups play prominent roles: the group PGL(2, R) of projective symmetries and the infinite-dimensional full group of diffeomorphisms Diff(RP1 ). We will see how these two types of symmetry interact. 2.1 Invariant differential operators on RP1 The language of invariant differential operators is an adequate language of differential geometry. The best known invariant differential operators are the de Rham differential of differential forms and the commutator of vector fields. These operators are invariant with respect to the action of the group of diffeomorphisms of the manifold. The expressions that describe these operations are independent of the choice of local coordinates. If a manifold M carries a geometric structure, the notion of the invariant differential operator changes accordingly: the full group of diffeomorphisms 29
- 44. 30 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE is restricted to the groups preserving the geometric structure. For instance, on a symplectic manifold M, one has the Poisson bracket, a binary invariant operation on the space of smooth functions, as well as the unitary operation assigning the Hamiltonian vector field to a smooth function. Another ex- ample, known to every student of calculus, is the divergence: the operator on a manifold with a fixed volume form assigning the function DivX to a vector field X. This operator is invariant with respect to volume preserving diffeomorphisms. Space of differential operators Dλ,µ(S1 ) Consider the space of linear differential operators on S1 from the space of λ-densities to the space of µ-densities A : Fλ(S1 ) → Fµ(S1 ) with arbitrary λ, µ ∈ R. This space will be denoted by Dλ,µ(S1) and its subspace of operators of order ≤ k by Dk λ,µ(S1). The space Dλ,µ(S1) is acted upon by Diff(S1); this action is as follows: Tλ,µ f (A) = Tµ f ◦ A ◦ Tλ f−1 , f ∈ Diff(S1 ) (2.1.1) where Tλ is the Diff(S1)-action on tensor densities (1.5.5). For any parameter x on S1, a k-th order differential operator is of the form A = ak(x) dk dxk + ak−1(x) dk−1 dxk−1 + · · · + a0(x), where ai(x) are smooth functions on S1. Exercise 2.1.1. Check that the expression σ(A) = ak(x)(dx)µ−λ−k does not depend on the choice of the parameter. The density σ(A) is called the principal symbol of A; it is a well-defined tensor density of degree µ−λ−k. The principal symbol provides a Diff(S1)- invariant projection σ : Dk λ,µ(S1 ) → Fµ−λ−k(S1 ). (2.1.2)
- 45. 2.1. INVARIANT DIFFERENTIAL OPERATORS ON RP1 31 Linear projectively invariant operators Our goal is to describe differential operators on RP1 , invariant under pro- jective transformations. In the one-dimensional case, there is only one type of tensors, namely tensor densities φ(x)(dx)λ. Recall that the space of such tensor densities is denoted by Fλ(RP1 ). A classical result of projective differential geometry is classification of projectively invariant linear differential operators A : Fλ(RP1 ) → Fµ(RP1 ) (see [28]). Theorem 2.1.2. The space of PGL(2, R)-invariant linear differential oper- ators on tensor densities is generated by the identity operator from Fλ(RP1 ) to Fλ(RP1 ) and the operators of degree k given, in an affine coordinate, by the formula Dk : φ(x)(dx) 1−k 2 7→ dkφ(x) dxk (dx) 1+k 2 . (2.1.3) Proof. The action of SL(2, R) is given, in an affine chart, by the formula x 7→ ax + b cx + d . (2.1.4) Exercise 2.1.3. Prove that the operators Dk are PGL(2, R)-invariant. The infinitesimal version of formula 2.1.4 gives the action of the Lie algebra sl(2, R). Exercise 2.1.4. a) Prove that the sl(2, R)-action on RP1 is generated by the three vector fields d dx , x d dx , x2 d dx . (2.1.5) b) Prove that the corresponding action on Fλ(RP1 ) is given by the following operators (the Lie derivatives): Lλ d dx = d dx , Lλ x d dx = x d dx + λ, Lλ x2 d dx = x2 d dx + 2λ x . (2.1.6) Consider now a differential operator A = ak(x) dk dxk + · · · + a0(x) from Fλ(RP1 ) to Fµ(RP1 ) and assume that A is SL(2, R)-invariant. This means that A ◦ Lλ X = Lµ X ◦ A
- 46. 32 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE for all X ∈ sl(2, R). Take X = d/dx to conclude that all the coefficients ai(x) of A are con- stants. Now take X = xd/dx: Lµ x d dx ◦ X ai di dxi = X ai di dxi ◦ Lλ x d dx . Using (2.1.6), it follows that ai(i + λ − µ) = 0 for all i. Hence all ai but one vanish, and µ = λ + k where k is the order of A. Finally, take X = x2 d dx . One has Lλ+k x2 d dx ◦ dk dxk = dk dxk ◦ Lλ x2 d dx . If k ≥ 1 then, using (2.1.6) once again, one deduces that 2λ = 1 − k, as claimed; if k = 0, then A is proportional to the identity. The operator D1 is just the differential of a function. This is the only operator invariant under the full group Diff(RP1 ). The operator D2 is a Sturm-Liouville operator already introduced in Section 1.3. Such an oper- ator determines a projective structure on RP1 . Not surprisingly, the pro- jective structure, corresponding to D2, is the standard projective structure whose symmetry group is PGL(2, R). The geometric meaning of the opera- tors Dk with k ≥ 3 will be discussed in the next section. Comment The classification problem of invariant differential operators was posed by Veblen in his talk at ICM in 1928. Many important results have been ob- tained since then. The only unitary invariant differential operator on tensor fields (and tensor densities) is the de Rham differential, cf. [114, 179]. 2.2 Curves in RPn and linear differential operators In Sections 1.3 and 1.4 we discussed the relations between non-degenerate curves and linear differential operators in dimensions 1 and 2. In this section we will extend this construction to the multi-dimensional case. Constructing differential operators from curves We associate a linear differential operator A = dn+1 dxn+1 + an−1(x) dn−1 dxn−1 + · · · + a1(x) d dx + a0(x) (2.2.1)
- 47. 2.2. CURVES IN RPN AND LINEAR DIFFERENTIAL OPERATORS33 with a non-degenerate parameterized curve γ(x) in RPn . Consider a lift Γ(x) of the curve γ(x) to Rn+1. Since γ is non-degenerate, the Wronski determinant W(x) = |Γ(x), Γ0 (x), . . . , Γ(n) (x)| does not vanish. Therefore the vector Γ(n+1) is a linear combination of Γ, Γ0, . . . , Γ(n), more precisely, Γ(n+1) (x) + n X i=0 ai(x)Γ(i) (x) = 0. This already gives us a differential operator depending, however, on the lift. Let us find a new, canonical, lift for which the Wronski determinant identically equals 1. Any lift of γ(x) is of the form α(x)Γ(x) for some non- vanishing function α(x). The condition on this function is |αΓ, (αΓ)0 , . . . , (αΓ)(n) | = 1, and hence α(x) = W(x)−1/(n+1) . (2.2.2) For this lift the coefficient an(x) in the preceding formula vanishes and the corresponding operator is of the form (2.2.1). This operator is uniquely defined by the curve γ(x). Exercise 2.2.1. Prove that two curves define the same operator (2.2.1) if and only if they are projectively equivalent. Hint. The “if” part follows from the uniqueness of the canonical lift of the projective curve. The “only if” part is more involved and is discussed throughout this section. Tensor meaning of the operator A and Diff(S1 )-action Let us discuss how the operator A depends on the parameterization of the curve γ(x). The group Diff(S1) acts on parameterized curves by reparam- eterization. To a parameterized curve we assigned a differential operator (2.2.1). Thus one has an action of Diff(S1) on the space of such operators. We call it the geometric action. Let us define another, algebraic action of Diff(S1) on the space of oper- ators (2.2.1) A 7→ T n+2 2 f ◦ A ◦ T − n 2 f−1 , f ∈ Diff(S1 ), (2.2.3)
- 48. 34 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE which is, of course, a particular case of the action of Diff(S1) on Dλ,µ(S1) as in (2.1.1). In other words, A ∈ Dn+1 − n 2 , n+2 2 (S1). Exercise 2.2.2. The action of Diff(S1) on Dλ,µ(S1) preserves the specific form of the operators (2.2.1), namely, the highest-order coefficient equals 1 and the next highest equals zero, if and only if λ = − n 2 and µ = n + 2 2 . Theorem 2.2.3. The two Diff(S1)-actions on the space of differential op- erators (2.2.1) coincide. Proof. Let us start with the geometric action. Consider a new parameter y = f(x) on γ. Then Γx = Γyf0 , Γxx = Γyy(f0 )2 + Γyf00 , etc., where Γ is a lifted curve and f0 denotes df/dx. It follows that |Γ, Γx, . . . , Γx...x| = |Γ, Γy, . . . , Γy...y|(f0 )n(n+1)/2 , and, therefore, the Wronski determinant W(x) is a tensor density of degree n(n + 1)/2, that is, an element of Fn(n+1)/2. Hence the coordinates of the canonical lift αΓ given by (2.2.2) are tensor densities of degree −n/2 (we already encountered a particular case n = 2 in Exercise 1.6.5). Being the coordinates of the canonical lift αΓ, the solutions of the equation Aφ = 0 (2.2.4) are −n/2-densities. From the very definition of the algebraic action (2.2.3) it follows that the kernel of the operator A consists of −n/2-densities. It remains to note that the kernel uniquely defines the corresponding operator. The brevity of the proof might be misleading. An adventurous reader may try to prove Theorem 2.2.3 by a direct computation. Even for the Sturm-Liouville (n = 2) case this is quite a challenge (see, e.g. [37]). Example 2.2.4. The SL(2, R)-invariant linear differential operator (2.1.3) fits into the present framework. This operator corresponds to a remarkable parameterized projective curve in RPk−1 , called the normal curve, uniquely characterized by the following property. The parameter x belongs to S1 and
- 49. 2.2. CURVES IN RPN AND LINEAR DIFFERENTIAL OPERATORS35 corresponds to the canonical projective structure on S1. If one changes the parameter by a fractional-linear transformation x 7→ (ax + b)/(cx + d), the resulting curve is projectively equivalent to the original one. In appropriate affine coordinates, this curve is given by γ = (1 : x : x2 : · · · : xk−1 ). (2.2.5) Dual operators and dual curves Given a linear differential operator A : Fλ → Fµ on S1, its dual operator A∗ : F1−µ → F1−λ is defined by the equality Z S1 A(φ)ψ = Z S1 φA∗ (ψ) for any φ ∈ Fλ and ψ ∈ F1−µ. The operation A 7→ A∗ is Diff(S1)-invariant. An explicit expression for the dual operator was already given (1.4.5). If λ + µ = 1 then the operator A∗ has the same domain and the same range as A. In this case, there is a decomposition A = A + A∗ 2 + A − A∗ 2 into the symmetric and skew-symmetric parts. Now let A ∈ Dn+1 − n 2 , n+2 2 (S1) be the differential operator (2.2.1) constructed from a projective curve γ(x). The modules F−n/2 and F(n+2)/2 are dual to each other. Therefore A can be decomposed into the symmetric and skew- symmetric parts, and this decomposition is independent of the choice of the parameter on the curve. This fact was substantially used in the proof of Theorem 1.4.3. Consider a projective curve γ(x) ⊂ RPn , its canonical lift Γ(x) ⊂ Rn+1 and the respective differential operator A. The coordinates of the curve Γ satisfy equation (2.2.4). These coordinates are linear functions on Rn+1. Thus the curve Γ lies in the space, dual to ker A, and so ker A is identified with Rn+1∗ . Now let us define a smooth parameterized curve Γ̃(x) in Rn+1∗ . Given a value of the parameter x, consider the solution φx of equation (2.2.4) satisfying the following n initial conditions: φx(x) = φ0 x(x) = . . . = φ(n−1) x (x) = 0; (2.2.6) such a solution is unique up to a multiplicative constant. The solution φx is a vector in Rn+1∗ , and we set: Γ̃(x) = φx. Define the projective curve γ̃(x) ⊂ RPn∗ as the projection of Γ̃(x).
- 50. 36 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE Exercise 2.2.5. Prove that the curve γ̃ coincides with the projectively dual curve γ∗. Dual curves correspond to dual differential operators We have two notions of duality, one for projective curves and one for differ- ential operators. The next classical result shows that the two agree. Theorem 2.2.6. Let A be the differential operator corresponding to a non- degenerate projective curve γ(x) ⊂ RPn . Then the differential operator, corresponding to the projectively dual curve γ∗(x), is (−1)n+1A∗. Proof. Let U = Ker A, V = Ker A∗. We will construct a non-degenerate pairing between these spaces. Let φ and ψ be −n/2-densities. The expression A(φ)ψ − φA∗ (ψ) (2.2.7) is a differential 1-form on S1. The integral of (2.2.7) vanishes, and hence there exists a function B(φ, ψ)(x) such that A(φ)ψ − φA∗ (ψ) = B0 (φ, ψ)dx. (2.2.8) If A is given by (2.2.1) then B(φ, ψ) = φ(n) ψ − φ(n−1) ψ0 + · · · + (−1)n φψ(n) + b(φ, ψ), where b is a bidifferential operator of degree ≤ n − 1. If φ ∈ U and ψ ∈ V then the left hand side of (2.2.8) vanishes, and therefore B(φ, ψ) is a constant. It follows that B determines a bilinear pairing of spaces U and V . The pairing B is non-degenerate. Indeed, fix a parameter value x = x0, and choose a special basis φ0, . . . , φn ∈ U such that φ (j) i (x0) = 0 for all i 6= j; i, j = 0, . . . , n, and φ (i) i (x0) = 1 for all i. Let ψi ∈ V be the basis in V defined similarly. In these bases, the matrix of B(φ, ψ)(x0) is triangular with the diagonal elements equal to ±1. The pairing B allows us to identify U∗ with V . Consider the curve Γ̃(x) associated with the operator A; this curve belongs to U and consists of solutions (2.2.6). Let b Γ(x) ⊂ V be a similar curve corresponding to A∗. We want to show that these two curves are dual with respect to the pairing B, that is, B(Γ̃(i) (x0), b Γ(x0)) = 0, i = 0, . . . , n − 1 (2.2.9)
- 51. 2.2. CURVES IN RPN AND LINEAR DIFFERENTIAL OPERATORS37 for all parameter values x0. Indeed, the vector Γ̃(i)(x0) belongs to the space of solutions U and this solution vanishes at x0. The function b Γ(x0) ∈ V vanishes at x0 with the first n − 1 derivatives, and (2.2.9) follows from the above expression for the operator B. Therefore γ̃∗ = b γ, that is, the curves corresponding to A and A∗ are projectively dual. Exercise 2.2.7. Prove the following explicit formula: B(φ, ψ) = X r+s+t≤n (−1)r+t+1 r + t r a (r) r+s+t+1φ(s) ψ(t) . Remark 2.2.8. If A is a symmetric operator, A∗ = A, then B is a non- degenerate skew-symmetric bilinear form, i.e., a symplectic structure, on the space Ker A – cf. [166]. Monodromy If γ is a closed curve, then the operator A has periodic coefficients. The converse is not at all true. Let A be an operator with periodic coefficients, in other words, a differential operator on S1. The solutions of the equation Aφ = 0 are not necessarily periodic; they are defined on R, viewed as the universal covering of S1 = R/2πZ. One obtains a linear map on the space of solutions: T : φ(x) 7→ φ(x + 2π) called the monodromy. Monodromy was already mentioned in Sections 1.3 and 1.6. Consider in more detail the case of operators (2.2.1). The Wronski de- terminant of any (n+1)-tuple of solutions is constant. This defines a volume form on the space of solutions. Since T preserves the Wronski determinant, the monodromy belongs to SL(n + 1, R). Note however that this element of SL(n + 1, R) is defined up to a conjugation, for there is no natural basis in ker A and there is no way to identify ker A with Rn+1; only a conjugacy class of T has an invariant meaning. Consider a projective curve γ(x) associated with a differential operator A on S1. Let Γ(x) ⊂ Rn+1 be the canonical lift of γ(x). Both curves are not necessarily closed, but satisfy the monodromy condition γ(x + 2π) = T(γ(x)), Γ(x + 2π) = T(Γ(x)), where T is a representative of a conjugacy class in SL(n + 1, R).
- 52. 38 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE As a consequence of Theorem 2.2.3, asserting the coincidence of the algebraic Diff(S1)-action (2.2.3) on the space of differential operators with the geometric action by reparameterization, we have the following statement. Corollary 2.2.9. The conjugacy class in SL(n + 1, R) of the monodromy of a differential operator (2.2.1) is invariant with respect to the Diff+(S1)- action, where Diff+(S1) is the connected component of Diff(S1). Comment Representation of parameterized non-degenerate curves in RPn (modulo equivalence) by linear differential operators was a basic idea of projective differential geometry of the second half of XIX-th century. We refer to Wilczynski’s book [231] for a first systematic account of this approach. Our proof of Theorem 2.2.6 follows that of [231] and [13]; a different proof can be found in [106]. 2.3 Homotopy classes of non-degenerate curves Differential operators on RP1 of the special form (2.2.1) correspond to non- degenerate curves in RPn . In this section we give a topological classi- fication of such curves. We study homotopy equivalence classes of non- degenerate immersed curves with respect to the homotopy, preserving the non-degeneracy. This allows us to distinguish interesting classes of curves, such as that of convex curves. Curves in S2 : a theorem of J. Little Let us start with the simplest case, the classification problem for non- degenerate curves on the 2-sphere. Figure 2.1: Non-degenerate curves on S2 Theorem 2.3.1. There are 3 homotopy classes of non-degenerate immersed non-oriented closed curves on S2 represented by the curves in figure 2.1.
- 53. 2.3. HOMOTOPY CLASSES OF NON-DEGENERATE CURVES 39 Proof. Recall the classical Whitney theorem on the regular homotopy clas- sification of closed plane immersed curves. To such a curve one assigns the winding number: a non-negative integer equal to the total number of turns of the tangent line (see figure 2.2). The curves are regularly homotopic if and only if their winding numbers are equal. The spherical version of the Whitney theorem is simpler: there are only 2 regular homotopy classes of closed immersed curves on S2, represented by the first and the second curves in figure 2.1. The complete invariant is the parity of the number of double points. n Figure 2.2: Winding number n The Whitney theorem extends to non-degenerate plane curves and the proof dramatically simplifies. Lemma 2.3.2. The winding number is the complete invariant of non-dege- nerate plane curves with respect to non-degenerate homotopy. Proof. A non-degenerate plane curve can be parameterized by the angle made by the tangent line with a fixed direction. In such a parameterization, a linear homotopy connects two curves with the same winding number. We are ready to proceed to the proof of Theorem 2.3.1. Part I. Let us prove that the three curves in figure 2.1 are not ho- motopic as non-degenerate curves. The second curve is not even regularly homotopic to the other two. We need to prove that the curves 1 and 3 are not homotopic. The curve 1 is convex: it intersects any great circle at at most two points. We understand intersections in the algebraic sense, that is, with multiplicities. For example, the curve y = x2 has double intersection with the x-axis. Lemma 2.3.3. A convex curve remains convex under homotopies of non- degenerate curves.
- 54. 40 CHAPTER 2. GEOMETRY OF PROJECTIVE LINE Proof. Arguing by contradiction, assume that there is a homotopy destroy- ing convexity. Convexity is an open condition. Consider the first moment when the curve fails to be convex. At this moment, there exists a great cir- cle intersecting the curve with total multiplicity four. The following 3 cases are possible: a) four distinct transverse intersections, b) two transverse and one tangency, c) two tangencies, see figure 2.3. Note that a non-degenerate curve cannot have intersection multiplicity 2 with a great circle at a point. a) b) c) Figure 2.3: Total multiplicity 4 Case a) is impossible: since transverse intersection is an open condition, this cannot be the first moment when the curve fails to be convex. In cases b) and c) one can perturb the great circle so that the intersections become transverse and we are back to case a). In case b) this is obvious, as well as in case c) if the two points of tangency are not antipodal, see figure 2.3. For antipodal points, one rotates the great circle about the axis connecting the tangency points. Part 2. Let us now prove that a non-degenerate curve on S2 is non- degenerate homotopic to a curve in figure 2.1. Unlike the planar case, non- degenerate curves with winding number n and n + 2, where n ≥ 2, are non-degenerate homotopic, see figure 2.4. The apparent inflection points are not really there; see [137] for a motion picture featuring front-and-back view. The authors recommend the reader to repeat their experience and to draw the picture on a well inflated ball. Therefore, any curve in figure 2.2 is, indeed, homotopic to a curve in figure 2.1 in the class of non-degenerate curves. Lemma 2.3.4. A non-degenerate curve on the 2-sphere is homotopic, in the class of non-degenerate curves, to a curve that lies in a hemisphere. Proof. If the curve is convex, then it already lies in a hemisphere. The proof of this fact is similar to that of Lemma 2.3.3. If the curve is not convex, then
- 55. Exploring the Variety of Random Documents with Different Content
- 56. Standards varied in size according to the rank of the person entitled to them. A MS. of the time of Henry VII. gives the following dimensions:—For that of the king, a length of eight yards; for a duke, seven; for an earl, six; a marquis, six and a half; a viscount, five and a half; a baron, five; a knight banneret, four and a half; and for a knight, four yards. In view of these figures one can easily realise the derivation of the word standard—a thing that is meant to stand; to be rather fastened in the ground as a rallying point than carried, like a banner, about the field of action. At the funeral of Nelson we find his banner of arms and standard borne in the procession, while around his coffin are the bannerolls, square banner-like flags bearing the various arms of his family lineage. We see these latter again in an old print of the funeral procession of General Monk, in 1670, and in a still older print of the burial of Sir Philip Sydney, four of his near kindred carrying by the coffin these indications of his descent. At the funeral of Queen Elizabeth we find six bannerolls of alliances on the paternal side and six on the maternal. The standard of Nelson bears his motto, Palmam qui meruit ferat, but instead of the Cross of St. George it has the union of the crosses of St. George, St. Andrew, and St. Patrick, since in 1806, the year of his funeral, the England of mediæval days had expanded into the Kingdom of Great Britain and Ireland. In the imposing funeral procession of the great Duke of Wellington we find again amongst the flags not only the national flag, regimental colours, and other insignia, but the ten bannerolls of the Duke's pedigree and descent, and his personal banner and standard. Richard, Earl of Salisbury, in the year 1458, ordered that at his interment there be banners, standards, and other accoutrements, according as was usual for a person of his degree and what was then held fitting, remains, in the case of State funerals, equally so at the present day. The Pennon is a small, narrow flag, forked or swallow-tailed at its extremity. This was carried on the lance. Our readers will recall the
- 57. knight in Marmion, who On high his forky pennon bore, Like swallow's tail in shape and hue. We read in the Roll of Karlaverok, as early as the year 1300, of Many a beautiful pennon fixed to a lance, And many a banner displayed; and of the knight in Chaucer's Canterbury Tales, we hear that By hys bannere borne is hys pennon Of golde full riche. The pennon bore the arms of the knight, and they were in the earlier days of chivalry so emblazoned upon it as to appear in their proper position not when the lance was held erect but when held horizontally for the charge. The earliest brass now extant, that of Sir John Daubernoun, at Stoke d'Abernon Church, in Surrey, represents the knight as bearing a lance with pennon. Its date is 1277, and the device is a golden chevron on a field of azure. In this example the pennon, instead of being forked, comes to a single point. The pennon was the ensign of those knights who were not bannerets, and the bearers of it were therefore sometimes called pennonciers; the term is derived from the Latin word for a feather, penna, from the narrow, elongated form. The pennons of our lancer regiments (Fig. 30) give one a good idea of the form, size, and general effect of the ancient knightly pennon, though they do not bear distinctive charges upon them, and thus fail in one notable essential to recall to our minds the brilliant blazonry and variety of device that must have been so marked and effective a feature when the knights of old took the field. In a drawing of the year 1813, of the Royal Horse Artillery, we find the men armed with lances, and these with pennons of blue and white, as we see in Fig. 31.[11]
- 58. Of the thirty-seven pennons borne on lances by various knights represented in the Bayeux tapestry, twenty-eight have triple points, while others have two, four, or five. The devices upon these pennons are very various and distinctive, though the date is before the period of the definite establishment of heraldry. Examples of these may be seen in Figs. 39, 40, 41, 42. The pennoncelle, or pencel, is a diminutive of the pennon, small as that itself is. Such flags were often supplied in large quantities at any special time of rejoicing or of mourning. At the burial in the year 1554 of the nobull Duke of Norffok, we note amongst other items a dosen of banerolles of ys progene, a standard, a baner of damaske, and xij dosen penselles. At the burial of Sir William Goring we find ther was viij dosen of penselles, while at the Lord Mayor's procession in 1555 we read that there were ij goodly pennes [State barges] deckt with flages and stremers and a m penselles. This m, or thousand, we can perhaps scarcely take literally, though in another instance we find the cordes were hanged with innumerable pencelles.[12] The statement of the cost of the funeral of Oliver Cromwell is interesting, as we see therein the divers kinds of flags that graced the ceremony. The total cost of the affair was over £28,000, and the unhappy undertaker, a Mr. Rolt, was paid very little, if any, of his bill. The items include six gret banners wrought on rich taffaty in oil, and gilt with fine gold, at £6 each. Five large standards, similarly wrought, at a cost of £10 each; six dozen pennons, a yard long, at a sovereign each; forty trumpet banners, at forty shillings apiece; thirty dozen of pennoncelles, a foot long, at twenty shillings a dozen; and twenty dozen ditto at twelve shillings the dozen. Poor Rolt! In the accompte and reckonyng for the Lord Mayor's Show of 1617 we find payde to Jacob Challoner, painter, for a greate square banner, the Prince's Armes, the somme of seven pounds. We also find, More to him for the new payntyng and guyldyng of ten trumpet banners, for payntyng and guyldyng of two long pennons of
- 59. the Lord Maior's armes on callicoe, and many other items that we need not set down, the total cost of the flag department being £67 15s. 10d., while for the Lord Mayor's Show of the year 1685 we find that the charge for this item was the handsome sum of £140. The Pennant, or pendant, is a long narrow flag with pointed end, and derives its name from the Latin word signifying to hang. Examples of it may be seen in Figs. 20, 21, 23, 24, 36, 38, 100, 101, 102, and 103, and some of the flags employed in ship-signalling are also of pennant form. It was in Tudor times called the streamer. Though such a flag may at times be found pressed into the service of city pageantry, it is more especially adapted for use at sea, since the lofty mast, the open space far removed from telegraph-wires, chimney-pots, and such-like hindrances to its free course, and the crisp sea-breeze to boldly extend it to its full length, are all essential to its due display. When we once begin to extend in length, it is evident that almost anything is possible: the pendant of a modern man-of-war is some twenty yards long, while its breadth is barely six inches, and it is evident that such a flag as that would scarcely get a fair chance in the general survival of the fittest in Cheapside. It is charged at the head with the Cross of St. George. Figs. 26, 27, 74 are Tudor examples of such pendants, while Fig. 140 is a portion at least of the pendant flown by colonial vessels on war service, while under the same necessarily abbreviated conditions may be seen in Fig. 151 the pendant of the United States Navy, in 157 that of Chili, and in 173 that of Brazil. In mediæval days many devices were introduced, the streamer being made of sufficient width to allow of their display. Thus Dugdale gives an account of the fitting up of the ship in which Beauchamp, fifth Earl of Warwick, during the reign of Henry VI., went over to France. The original bill between this nobleman and William Seburgh, citizen and payntour of London, is still extant, and we see from it that amongst other things provided was the grete stremour for the shippe xl yardes in length and viij yardes in brede. These noble dimensions gave ample room for display of the
- 60. badge of the Warwicks,[13] so we find it at the head adorned with a grete bere holding a ragged staffe, and the rest of its length powdrid full of raggid staves, A stately ship, With all her bravery on, and tackle trim, Sails filled, and streamers waving. Machyn tells us in his diary for August 3rd, 1553, how The Queen came riding to London, and so on to the Tower, makyng her entry at Aldgate, and a grett nombur of stremars hanging about the sayd gate, and all the strett unto Leydenhalle and unto the Tower were layd with graffel, and all the crafts of London stood with their banars and stremars hangyd over their heds. In the picture by Volpe in the collection at Hampton Court of the Embarkation of Henry VIII. from Dover in the year 1520, to meet Francis I. at the Field of the Cloth of Gold, we find, very naturally, a great variety and display of flags of all kinds. Figs. 20, 21, 23 are streamers therein depicted, the portcullis, Tudor rose, and fleur-de-lys being devices of the English king, while the particular ground upon which they are displayed is in each case made up of green and white, the Tudor livery colours. We may see these again in Fig. 71, where the national flag of the Cross of St. George has its white field barred with the Tudor green. In the year 1554 even the naval uniform of England was white and green, both for officers and mariners, and the City trained bands had white coats welted with green. Queen Elizabeth, though of the Tudor race, took scarlet and black as her livery colours; the House of Plantaganet white and red; of York, murrey and blue; of Lancaster, white and blue; of Stuart, red and yellow. The great nobles each also had their special liveries; thus in a grand review of troops on Blackheath, on May 16th, 1552, we find that the Yerle of Pembroke and ys men of armes had cotes blake bordered with whyt, while the retainers of the Lord Chamberlain were in red and white, those of the Earl of Huntingdon in blue, and so forth.
- 61. In the description of one of the City pageants in honour of Henry VII. we find among the baggs (i.e., badges), a rede rose and a wyght in his mydell, golde floures de luces, and portcullis also in golde, the wallys of the Pavilion whereon these were displayed being chekkyrs of whyte and grene. The only other flag form to which we need make any very definite reference is the Guidon. The word is derived from the French guide- homme, but in the lax spelling of mediæval days it undergoes many perversions, such as guydhome, guydon, gytton, geton, and such- like more or less barbarous renderings. Guidon is the regulation name now applied to the small standards borne by the squadrons of some of our cavalry regiments. The Queen's guidon is borne by the first squadron; this is always of crimson silk; the others are the colour of the regimental facings. The modern cavalry guidon is square in form, and richly embroidered, fringed, and tasselled. A mediæval writer on the subject lays down the law that a guydhome must be two and a half yardes or three yardes longe, and therein shall be no armes putt, but only the man's crest, cognizance, and device, and from that, from his standard or streamer a man may flee; but not from his banner or pennon bearinge his armes. The guidon is largely employed at State or ceremonious funeral processions; we see it borne, for instance, in the illustrations of the funeral of Monk in 1670, of Nelson in 1806, of Wellington in 1852. In all these cases it is rounded in form, as in Fig. 28. Like the standard, the guidon bears motto and device, but it is smaller, and has not the elongated form, nor does it bear the Cross of St. George. In divers countries and periods very diverse forms may be encountered, and to these various names have been assigned, but it is needless to pursue their investigation at any length, as in some cases the forms are quite obsolete; in other cases, while its form is known to us its name is lost, while in yet other instances we have various old names of flags mentioned by the chroniclers and poets to which we are unable now to assign any very definite notion of their form. In some cases, again, the form we encounter may be of some
- 62. eccentric individuality that no man ever saw before, or ever wants to see again, or, as in Fig. 33, so slightly divergent from ordinary type as to scarcely need a distinctive name. One of the flags represented in the Bayeux tapestry is semi-circular. Fig. 32 defies classification, unless we regard it as a pennon that, by snipping, has travelled three-quarters of the way towards being a banner. Fig. 35, sketched from a MS. of the early part of the fourteenth century, in the British Museum, is of somewhat curious and abnormal form. It is of religious type, and bears the Agnus Dei. The original is in a letter of Philippe de Mezières, pleading for peace and friendship between Charles VI. of France and Richard II. of England. Flags are nowadays ordinarily made of bunting, a woollen fabric which, from the nature of its texture and its great toughness and durability, is particularly fitted to stand wear and tear. It comes from the Yorkshire mills in pieces of forty yards in length, while the width varies from four to thirty-six inches. Flags are only printed when of small size, and when a sufficient number will be required to justify the expense of cutting the blocks. Silk is also used, but only for special purposes. Flag-devising is really a branch of heraldry, and should be in accordance with its laws, both in the forms and the colours introduced. Yellow in blazonry is the equivalent of gold, and white of silver, and it is one of the requirements of heraldry that colour should not be placed upon colour, nor metal on metal. Hence the red and blue in the French tricolour (Fig. 191) are separated by white; the black and red of Belgium (Fig. 236) by yellow. Such unfortunate combinations as the yellow, blue, red, of Venezuela (Fig. 170); the yellow, red, green of Bolivia (Fig. 171); the red and blue of Hayti (Fig. 178); the white and yellow of Guatemala (Fig. 162), are violations of the rule in countries far removed from the influence of heraldic law. This latter instance is a peculiarly interesting one; it is the flag of Guatemala in 1851, while in 1858 this was changed to that represented in Fig. 163. In the first case the red and the blue are in contact, and the white and the yellow; while in the second the
- 63. same colours are introduced, but with due regard to heraldic law, and certainly with far more pleasing effect. One sees the same obedience to this rule in the special flags used for signalling, where great clearness of definition at considerable distances is an essential. Such combinations as blue and black, red and blue, yellow and white, carry their own condemnation with them, as anyone may test by actual experiment; stripes of red and blue, for instance, at a little distance blending into purple, while white and yellow are too much alike in strength, and when the yellow has become a little faded and the white a little dingy they appear almost identical. We have this latter combination in Fig. 198, the flag of the now vanished Papal States. It is a very uncommon juxtaposition, and only occurs in this case from a special religious symbolism into which we need not here enter. The alternate red and green stripes in Fig. 63 are another violation of the rule, and have a very confusing effect.[14] The colours of by far the greatest frequency of occurrence are red, white, and blue; yellow also is not uncommon; orange is only found once, in Fig. 249, where it has a special significance, since this is the flag of the Orange Free State. Green occurs sparingly. Italy (Fig. 197) is perhaps the best known example. We also find it in the Brazilian flag (Fig. 169), the Mexican (Fig. 172), in the Hungarian tricolor (Fig. 214), and in Figs. 199, 201, 209, the flags of smaller German States, but it is more especially associated with Mohammedan States, as in Figs. 58, 63, 64, 235. Black is found but seldom, but as heraldic requirements necessitate that it should be combined either with white or yellow, it is, when seen, exceptionally brilliant and effective. We see it, for example, in the Royal Standard of Spain, (Fig. 194), in Figs. 207 and 208, flags of the German Empire, in Fig. 226, the Imperial Standard of Russia, and in Fig. 236, the brilliant tricolor of the Belgians.[15] In orthodox flags anything of the nature of an inscription is very seldom seen. We find a reference to order and progress on the
- 64. Brazilian flag (Fig. 169), while the Turkish Imperial Standard (Fig. 238) bears on its scarlet folds the monogram of the Sultan; but these exceptions are rare.[16] We have seen that, on the contrary, on the flags of insurgents and malcontents the inscription often counts for much. On the alteration of the style in the year 1752 this necessary change was made the subject of much ignorant reproach of the government of the day, and was used as a weapon of party warfare. An amusing instance of this feeling occurs in the first plate of Hogarth's election series, where a malcontent, or perhaps only a man anxious to earn a shilling, carries a big flag inscribed, Give us back our eleven days. The flags of the Covenanters often bore mottoes or texts. Fig. 34 is a curious example: the flag hoisted by the crew of H.M.S. Niger when they opposed the mutineers in 1797 at Sheerness. It is preserved in the Royal United Service Museum. It is, as we have seen, ordinarily the insubordinate and rebellious who break out into inscriptions of more or less piety or pungency, but we may conclude that the loyal sailors fighting under the royal flag adopted this device in addition as one means the more of fighting the rebels with their own weapons. During the Civil War between the Royalists and Parliamentarians, we find a great use made of flags inscribed with mottoes. Thus, on one we see five hands stretching at a crown defended by an armed hand issuing from a cloud, and the motto, Reddite Cæsari. In another we see an angel with a flaming sword treading a dragon underfoot, and the motto, Quis ut Deus, while yet another is inscribed, Courage pour la Cause. On a fourth we find an ermine, and the motto, Malo mori quam fœdari—It is better to die than to be sullied, in allusion to the old belief that the ermine would die rather than soil its fur. Hence it is the emblem of purity and stainless honour. The blood-red flag is the symbol of mutiny and of revolution. As a sign of disaffection it was twice, at the end of last century, displayed in the Royal Navy. A mutiny broke out at Portsmouth in April, 1797, for an advance of pay; an Act of Parliament was passed to sanction
- 65. the increase of expenditure, and all who were concerned in it received the royal pardon, but in June of the same year, at Sheerness, the spirit of disaffection broke out afresh, and on its suppression the ringleaders were executed. It is characteristic that, aggrieved as these seamen were against the authorities, when the King's birthday came round, on June 4th, though the mutiny was then at its height, the red flags were lowered, the vessels gaily dressed in the regulation bunting, and a royal salute was fired. Having thus demonstrated their real loyalty to their sovereign, the red flags were re-hoisted, and the dispute with the Admiralty resumed in all its bitterness. The white flag is the symbol of amity and of good will; of truce amidst strife, and of surrender when the cause is lost. The yellow flag betokens infectious illness, and is displayed when there is cholera, yellow fever, or such like dangerous malady on board ship, and it is also hoisted on quarantine stations. The black flag signifies mourning and death; one of its best known uses in these later days is to serve as an indication after an execution that the requirements of the law have been duly carried out. Honour and respect are expressed by dipping the flag. At any parade of troops before the sovereign the regimental flags are lowered as they pass the saluting point, and at sea the colours are dipped by hauling them smartly down from the mast-head and then promptly replacing them. They must not be suffered to remain at all stationary when lowered, as a flag flying half-mast high is a sign of mourning for death, for defeat, or for some other national loss, and it is scarcely a mark of honour or respect to imply that the arrival of the distinguished person is a cause of grief or matter for regret. In time of peace it is an insult to hoist the flag of one friendly nation above another, so that each flag must be flown from its own staff. Even as early as the reign of Alfred England claimed the sovereignty of the seas. Edward III. is more identified with our early naval glories than any other English king; he was styled King of the
- 66. Seas, a name of which he appears to have been very proud, and in his coinage of gold nobles he represented himself with shield and sword, and standing in a ship full royally apparelled. He fought on the seas under many disadvantages of numbers and ships: in one instance until his ship sank under him, and at all times as a gallant Englishman. If any commander of an English vessel met the ship of a foreigner, and the latter refused to salute the English flag, it was enacted that such ship, if taken, was the lawful prize of the captain. A very notable example of this punctilious insistance on the respect to the flag arose in May, 1554, when a Spanish fleet of one hundred and sixty sail, escorting the King on his way to England to his marriage with Queen Mary, fell in with the English fleet under the command of Lord Howard, Lord High Admiral. Philip would have passed the English fleet without paying the customary honours, but the signal was at once made by Howard for his twenty-eight ships to prepare for action, and a round shot crashed into the side of the vessel of the Spanish Admiral. The hint was promptly taken, and the whole Spanish fleet struck their colours as homage to the English flag. In the year 1635 the combined fleets of France and Holland determined to dispute this claim of Great Britain, but on announcing their intention of doing so an English fleet was at once dispatched, whereupon they returned to their ports and decided that discretion was preferable even to valour. In 1654, on the conclusion of peace between England and Holland, the Dutch consented to acknowledge the English supremacy of the seas, the article in the treaty declaring that the ships of the Dutch—as well ships of war as others— meeting any of the ships of war of the English, in the British seas, shall strike their flags and lower their topsails in such manner as hath ever been at any time heretofore practised. After another period of conflict it was again formally yielded by the Dutch in 1673. Political changes are responsible for many variations in flags, and the wear and tear of Time soon renders many of the devices obsolete. On turning, for instance, to Nories' Maritime Flags of all Nations, a
- 67. little book published in 1848, many of the flags are at once seen to be now out of date. The particular year was one of exceptional political agitation, and the author evidently felt that his work was almost old-fashioned even on its issue. The accompanying illustrations, he says, having been completed prior to the recent revolutionary movements on the Continent of Europe, it has been deemed expedient to issue the plate in its present state, rather than adopt the various tri-coloured flags, which cannot be regarded as permanently established in the present unsettled state of political affairs. The Russian American Company's flag, Fig. 59, that of the States of the Church, of the Kingdom of Sardinia, the Turkish Imperial Standard, Fig. 64, and many others that he gives, are all now superseded. For Venice he gives two flags, that for war and that for the merchant service. In each case the flag is scarlet, having a broad band of blue, which we may take to typify the sea, near its lower edge. From this rises in gold the winged lion of St. Mark, having in the war ensign a sword in his right paw, and in the peaceful colours of commerce a cross. Of thirty-five flags of all nations, given as a supplement to the Illustrated London News in 1858, we note that eleven are now obsolete: the East India Company, for instance, being now extinct, the Ionian Islands ceded to Greece, Tuscany and Naples absorbed into Italy, and so forth. In Figs. 52 and 53 we have examples of early Spanish flags, and in 54 and 55 of Portuguese, each and all being taken from a very quaint map of the year 1502. This map may be said to be practically the countries lying round the Atlantic Ocean, giving a good slice of Africa, a portion of the Mediterranean basin, the British Isles, most of South America, a little of North America, the West Indies,[17] etc., the object of the map being to show the division that Pope Alexander VI. kindly made between those faithful daughters of the Church—Spain and Portugal—of all the unclaimed portions of the world. Figs. 52 and 53 are types of flags flying on various Spanish possessions, while Figs. 54 and 55 are placed at different points on the map where Portugal held sway. On one place in Africa we see that No. 54 is surmounted by a white flag bearing the Cross of St.
- 68. George, so we may conclude that—Pope Alexander notwithstanding —England captured it from the Portuguese. At one African town we see the black men dancing round the Portuguese flag, while a little way off three of their brethren are hanging on a gallows, showing that civilization had set in with considerable severity there. The next illustration on this plate (Fig. 56) is taken from a sheet of flags published in 1735; it represents the Guiny Company's Ensign, a trading company, like the East India, Fig. 57, now no longer in existence. Fig. 62 is the flag of Savoy, an ancient sovereignty that, within the memory of many of our readers, has expanded into the kingdom of Italy. The break up of the Napoleonic régime in France, the crushing out of the Confederate States in North America, the dismissal from the throne of the Emperor of Brazil, have all, within comparatively recent years, led to the superannuation and disestablishment of a goodly number of flags and their final disappearance. We propose now to deal with the flags of the various nationalities, commencing, naturally, with those of our own country. We were told by a government official that the Universal Code of signals issued by England had led to a good deal of heartburning, as it is prefaced by a plate of the various national flags, the Union Flag of Great Britain and Ireland being placed first. But until some means can be devised by which each nationality can head the list, some sort of precedence seems inevitable. At first sight it seems as though susceptibilities might be saved by adopting an alphabetical arrangement, but this is soon found to be a mistake, as it places such powerful States as Russia and the United States nearly at the bottom of the list. A writer, Von Rosenfeld, who published a book on flags in Vienna in 1853, very naturally adopted this arrangement, but the calls of patriotism would not even then allow him to be quite consistent, since he places his material as follows:—Austria, Annam, Argentine, Belgium, Bolivia, and so forth, where it is evident Annam should lead the world and Austria be content to come in third. Apart from the difficulty of asking Spain, for instance, to admit that Bulgaria was so much in front of her, or to expect Japan to allow China so great a
- 69. precedence as the alphabetical arrangement favours, a second obstacle is found in the fact that the names of these various States as we Englishmen know them are not in many cases those by which they know themselves or are known by others. Thus a Frenchman would be quite content with the alphabetical arrangement that in English places his beloved country before Germany, but the Teuton would at once claim precedence, declaring that Deutschland must come before la belle France, and the Espagnol would not see why he should be banished to the back row just because we choose to call him a Spaniard. In the meantime, pending the Millenium, the flag that more than three hundred millions of people, the wide world over, look up to as the symbol of justice and liberty, will serve very well as a starting point, and then the great Daughter across the Western Ocean, that sprung from the Old Home, shall claim a worthy place next in our regard. The Continent of Europe must clearly come next, and such American nationalities as lie outside the United States, together with Asia and Africa, will bring up the rear. CHAPTER II. The Royal Standard—the Three Lions of England—the Lion Rampant of Scotland—Scottish sensitiveness as to precedence—the Scottish Tressure— the Harp of Ireland—Early Irish Flags—Brian Boru—the Royal Standards from Richard I. to Victoria—Claim to the Fleurs-de-Lys of France—Quartering Hanover—the Union Flag—St. George for England—War Cry—Observance of St. George's Day—the Cross of St. George—Early Naval Flags—the London Trained Bands—the Cross of St. Andrew—the Blue Blanket—Flags of the Covenanters—Relics of St. Andrew—Union of England and Scotland—the First Union Flag—Importance of accuracy in representations of it—the Union Jack —Flags of the Commonwealth and Protectorate—Union of Great Britain and Ireland—the Cross of St. Patrick—Labours of St. Patrick in Ireland— Proclamation of George III. as to Flags, etc.—the Second Union Flag— Heraldic Difficulties in its Construction—Suggestions by Critics—Regulations as to Fortress Flags—the White Ensign of the Royal Navy—Saluting the Flag— the Navy the Safeguard of Britain—the Blue Ensign—the Royal Naval Reserve —the Red Ensign of the Mercantile Marine—Value of Flag-lore.
- 70. Foremost amongst the flags of the British Empire the Royal Standard takes its position as the symbol of the tie that unites all into one great State. Its glowing blazonry of blue and scarlet and gold is brought before us in Fig. 44. The three golden lions on the scarlet ground are the device of England, the golden harp on the azure field is the device of Ireland, while the ruddy lion rampant on the field of gold[18] stands for Scotland. It may perhaps appear to some of our readers that the standard of the Empire should not be confined to such narrow limits; that the great Dominion of Canada, India, Australia, the ever-growing South Africa, might justly claim a place. Precedent, too, might be urged, since in previous reigns, Nassau, Hanover, and other States have found a resting-place in its folds, and there is much to be said in favour of a wider representation of the greater component parts of our world-wide Empire; but two great practical difficulties arise: the first is that the grand simplicity of the flag would be lost if eight or ten different devices were substituted for the three; and secondly, it would very possibly give rise to a good deal of jealousy and ill-feeling, since it would be impossible to introduce all. As it at present stands, it represents the central home of the Empire, the little historic seed-plot from whence all else has sprung, and to which all turn their eyes as the centre of the national life. All equally agree to venerate the dear mother land, but it is perhaps a little too much to expect that the people of Jamaica or Hong Kong would feel the same veneration for the beaver and maple-leaves of Canada, the golden Sun of India, or the Southern Cross of Australasia. As it must clearly be all or none, it seems that only one solution of the problem, the present one, is possible. In the same way the Union flag (Fig. 90) is literally but the symbol of England, Scotland, and Ireland, but far and away outside its primary significance, it floats on every sea the emblem of that Greater Britain in which all its sons have equal pride, and where all share equal honour as brethren of one family. The earliest Royal Standard bore but the three lions of England, and we shall see presently that in different reigns various modifications of its blazonry arose, either the result of conquest or of dynastic
- 71. possessions. Thus Figs. 43 and 44, though they bear a superficial likeness, tell a very different story; the first of these, that of George III., laying claim in its fourth quartering to lordship over Hanover and other German States, and in its second quarter to the entirely shadowy and obsolete claim over France, as typified by the golden fleurs-de-lys on the field of azure. How the three lions of England arose is by no means clear. Two lions were assigned as the arms of William the Conqueror, but there is no real evidence that he bore them. Heraldry had not then become a definite science, and when it did a custom sprang up of assigning to those who lived and died before its birth certain arms, the kindly theory being that such persons, had they been then living, would undoubtedly have borne arms, and that it was hard, therefore, that the mere accident of being born a hundred years too soon should debar them from possessing such recognition of their rank. Even so late as Henry II. the bearing is still traditional, and it is said that on his marriage with Alianore, eldest daughter of William, Duke of Aquitaine and Guienne, he incorporated with his own two lions the single lion that (it is asserted) was the device of his father-in-law. All this, however, is theory and surmise, and we do not really find ourselves on the solid ground of fact until we come to the reign of Richard Cœur-de-Lion. Upon his second Great Seal we have the three lions just as they are represented in Figs. 22, 43, 44, and as they have been borne for centuries by successive sovereigns on their arms, standards, and coinage, and as our readers may see them this day on the Royal Standard and on much of the money they may take out of their pockets. The date of this Great Seal of King Richard is 1195 A.D., so we have, at all events, a period of over seven hundred years, waiving a break during the Commonwealth, in which the three golden lions on their scarlet field have typified the might of England. The rampant lion within the tressure, the device of Scotland—seen in the second quarter of our Royal Standard, Fig. 44—is first seen on the Great Seal of King Alexander II., about A.D. 1230, and the same device, without any modification of colour or form[19] was borne by
- 72. all the Sovereigns of Scotland, and on the accession of James to the throne of the United Kingdom, in the year 1603, the ruddy lion ramping on the field of gold became an integral part of the Standard. The Scotch took considerable umbrage at their lion being placed in the second place, while the lions of England were placed first, as they asserted that Scotland was a more ancient kingdom than England, and that in any case, on the death of Queen Elizabeth of England, the Scottish monarch virtually annexed the Southern Kingdom to his own, and kindly undertook to get the Southerners out of a dynastic difficulty by looking after the interests of England as well as ruling Scotland. This feeling of jealousy was so bitter and so potent that for many years after the Union, on all seals peculiar to Scottish business and on the flags displayed north of the Tweed, the arms of Scotland were placed in the first quarter. It was also made a subject of complaint that in the Union Flag the cross of St. George is placed over that of St. Andrew (see Figs. 90, 91, 92), and that the lion of England acted as the dexter support of the royal shield instead of giving place to the Scottish Unicorn. One can only be thankful that Irish patriots have been too sensible or too indifferent to insist upon yet another modification, requiring that whensoever and wheresoever the Royal Standard be hoisted in the Emerald Isle the Irish harp should be placed in the first quarter. While it is clearly impossible to place the device of each nationality first, it is very desirable and, in fact, essential, that the National Arms and the Royal Standard should be identical in arrangement in all parts of the kingdom. The notion of unity would be very inadequately carried out if we had a London version for Buckingham Palace, an Edinburgh version for Holyrood, and presently found the Isle of Saints and gallant little Wales insisting on two other variants, and the Isle of Man in insurrection because it was not allowed precedence of all four. Even so lately as the year 1853, on the issue of the florin, the old jealousy blazed up again. A statement was drawn up and presented
- 73. to Lord Lyon, King of Arms, setting forth anew the old grievances of the lions in the Standard and the crosses in the Flag of the Union, and adding that the new two-shilling piece, called a florin, which has lately been issued, bears upon the reverse four crowned shields, the first or uppermost being the three lions passant of England; the second, or right hand proper, the harp of Ireland; the third, or left hand proper, the lion rampant of Scotland; the fourth, or lower, the three lions of England repeated. Your petitioners beg to direct your Lordship's attention to the position occupied by the arms of Scotland upon this coin, which are placed in the third shield instead of the second, a preference being given to the arms of Ireland over those of this kingdom. It is curious that this document tacitly drops claim to the first place. Probably most of our readers—Scotch, Irish, or English—feel but little sense of grievance in the matter, and are quite willing, if the coin be an insult, to pocket it. The border surrounding the lion is heraldically known as the tressure. The date and the cause of its introduction are lost in antiquity. The mythical story is that it was added by Achaius, King of Scotland, in the year 792, in token of alliance with Charlemagne, but in all probability these princes scarcely knew of the existence of each other. The French and the Scotch have often been in alliance, and there can be little doubt but that the fleurs-de-lys that adorn the tressure point to some such early association of the two peoples; an ancient writer, Nisbet, takes the same view, as he affirms that the Tressure fleurie encompasses the lyon of Scotland to show that he should defend the Flower-de-luses, and these to continue a defence to the lyon. The first authentic illustration of the tressure in the arms of Scotland dates from the year 1260. In the reign of James III., in the year 1471 it was ordaint that in tyme to cum thar suld be na double tresor about his armys, but that he suld ber armys of the lyoun, without ony mur. If this ever took effect it must have been for a very short time. We have seen no example of it. Ireland joined England and Scotland in political union on January 1st, 1801, but its device—the harp—was placed on the standard
- 74. centuries before by right of conquest. The first known suggestion for a real union on equal terms was made in the year 1642 in a pamphlet entitled The Generall Junto, or the Councell of Union; chosen equally out of England, Scotland, and Ireland for the better compacting of these nations into one monarchy. By H. P. This H. P. was one Henry Parker. Fifty copies only of this tract were issued, and those entirely for private circulation. To persuade to union and commend the benefit of it—says the author—will be unnecessary. Divide et impera (divide and rule) is a fit saying for one who aims at the dissipation and perdition of his country. Honest counsellors have ever given contrary advice. England and Ireland are inseparably knit; no severance is possible but such as shall be violent and injurious. Ireland is an integral member of the Kingdom of England: both kingdoms are coinvested and connexed, not more undivided than Wales or Cornwall. The conquest of Ireland was entered upon in the year 1172, in the reign of Henry II., but was scarcely completed until the surrender of Limerick in 1691. Until 1542 it was styled not the Kingdom but the Lordship of Ireland. An early standard of Ireland has three golden crowns on a blue field, and arranged over each other as we see the English lions placed; and a commission appointed in the reign of Edward IV., to enquire what really were the arms of Ireland, reported in favour of the three crowns. The early Irish coinage bears these three crowns upon it, as on the coins of Henry V. and his successors. Henry VIII. substituted the harp on the coins, but neither crowns nor harps nor any other device for Ireland appear in the Royal Standard until the year 1603, after which date the harp has remained in continuous use till the present day. In the Harleian MS., No. 304 in the British Museum, we find the statement that the armes of Irland is Gules iij old harpes gold, stringed argent (as in Fig. 87), and on the silver coinage for Ireland of Queen Elizabeth the shield bears these three harps. At her funeral Ireland was represented by a blue flag having a crowned harp of
- 75. gold upon it, and James I. adopted this, but without the crown, as a quartering in his standard: its first appearance on the Royal Standard of England. Why Henry VIII. substituted the harp for the three crowns is not really known. Some would have us believe that the king was apprehensive that the three crowns might be taken as symbolising the triple crown of the Pope; while others suggest that Henry, being presented by the Pope with the supposed harp of Brian Boru, was induced to change the arms of Ireland by placing on her coins the representation of this relic of her most celebrated native king. The Earl of Northampton, writing in the reign of James I., suggests yet a third explanation. The best reason, saith he, that I can observe for the bearing thereof is, it resembles that country in being such an instrument that it requires more cost to keep it in tune than it is worth.[20] The Royal Standard should only be hoisted when the Sovereign or some member of the royal family is actually within the palace or castle, or at the saluting point, or on board the vessel where we see it flying, though this rule is by no means observed in practice. The only exception really permitted to this is that on certain royal anniversaries it is hoisted at some few fortresses at home and abroad that are specified in the Queen's Regulations. The Royal Standard of England was, we have seen, in its earliest form a scarlet flag, having three golden lions upon it, and it was so borne by Richard I., John, Henry III., Edward I., and Edward II. Edward III. also bore it for the first thirteen years of his reign, so that this simple but beautiful flag was the royal banner for over one hundred and fifty years. Edward III., on his claim in the year 1340 to be King of France as well as of England, quartered the golden fleurs- de-lys of that kingdom with the lions of England.[21] This remained the Royal Standard throughout the rest of his long reign. Throughout the reign of Richard II. (1377 to 1399) the royal banner was divided in half by an upright line, all on the outer half being like that of
- 76. Edward III., while the half next the staff was the golden cross and martlets on the blue ground, assigned to Edward the Confessor, his patron saint, as shown in Fig. 19. On the accession of Henry IV. to the throne, the cross and martlets disappeared, and he reverted to the simple quartering of France and England. Originally the fleurs-de-lys were scattered freely over the field, semée or sown, as it is termed heraldically, so that besides several in the centre that showed their complete form, others at the margin were more or less imperfect. On turning to Fig. 188, an early French flag, we see this disposition of them very clearly. Charles V. of France in the year 1365 reduced the number to three, as in Fig. 184, whereupon Henry IV. of England followed suit; his Royal Standard is shown in Fig. 22. This remained the Royal Standard throughout the reigns of Henry V., Henry VI., Edward IV., Edward V., Richard III., Henry VII., Henry VIII., Edward VI., Mary and Elizabeth—a period of two hundred years. On the accession of the House of Stuart, the flag was rearranged. Its first and fourth quarters were themselves quartered again, these small quarterings being the French fleur-de-lys and the English lions; while the second quarter was the lion of Scotland, and the third the Irish harp; the first appearance of either of these latter kingdoms in the Royal Standard. This form remained in use throughout the reigns of James I., Charles I., Charles II., and James II. The last semblance of dominion in France had long since passed away, but it will be seen that alike on coinage, arms, and Standard the fiction was preserved, and Londoners may see at Whitehall the statue still standing of James II., bearing on its pedestal the inscription —Jacobus secundus Dei Gratia Angliæ, Scotiæ, Franciæ et Hiberniæ Rex. During the Protectorate, both the Union Flag and the Standard underwent several modifications, but the form that the personal Standard of Cromwell finally assumed may be seen in Fig. 83, where the Cross of St. George for England, St. Andrew for Scotland, and the harp for Ireland, symbolise the three kingdoms, while over all,
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