Operator’s Differential geometry with Riemannian Manifolds

International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 5 Issue 3|| March 2016 || PP.28-43
www.ijesi.org 28 | Page
Operator’s Differential geometry with Riemannian Manifolds
Dr. Mohamed M.Osman
Department of mathematics faculty of science University of Al-Baha – Kingdom of Saudi Arabia
ABSTRACT : In this paper some fundamental theorems , operators differential geometry – with operator
Riemannian geometry to pervious of differentiable manifolds which are used in an essential way in basic
concepts of Spectrum of Discrete , bounded Riemannian geometry, we study the defections, examples of the
problem of differentially projection mapping parameterization system on dimensional manifolds .
Keywords -Basic notions on differential geometry-The spectral geometry –The geometric global minima and
maxima-The geometric of Laplace and Dirac spinner Bounded – Heat trace Asymptotic closed manifolds –
Riemannian manifolds with same spectrum Bounded Harmonic function – compact Riemannian manifolds –
computations of spectrum.
I. Introduction
Differential forms and the exterior derivative provide one piece of analysis on manifolds which, as we have seen
, links in with global topological questions . There is much more on can do when on introduces a Riemannian
metric. Since the whole subject of Riemannian geometry is a huge to the use of differential forms. The study of
harmonic from and of geodesics in particular, we ignore completely hare questions related to curvature. The
spectrum does not in general determine the geometry of a manifold Nevertheless earthiness, some geometric
information can be extracted from the spectrum. In what follows, we define a spectral invariant to be anything
that is completely determined by the spectrum .A Riemannian manifold is a pair ).( gM consisting of a
smooth manifold M and a metric g on the tangent bundle, i.e. a smooth symmetric positive definite tensor field
on M . The tensor g is called a Riemannian metric on M
II. Basic Notions On Differential Geometry
2.1 Basic on topological Manifold
Definition 2.1.1 Topological Manifold
A topological manifold M of dimension n , is a topological space with the following properties:
(a) M Is a Hausdorff space . For ever pair of points Mgp , , there are disjoint open subsets MVU , such that
Up  and Vg  .
(b) M Is second countable. There exists accountable basis for the topology of M . (c) M is locally Euclidean of
dimension n . Every point of M has a neighborhood that is homeomorphism to an open subset of n
R .
Definition 2.1.3
A topological space M is called an m-dimensional topological manifold with boundary MM  if the
following conditions : (i) M is Hausdorff space.(ii) for any point Mp  there exists a neighborhood U of p
which is homeomorphism to an open subset m
HV  .(iii) M has a countable basis of open sets , can be
rephrased as follows any point Up  is contained in neighborhood U to mm
HD  the set M is a locally
homeomorphism to m
R or m
H the boundary MM  is subset of M which consists of points p .
Definition 2.1.4
Let X be a set a topology U for X is collection of X satisfying : (i)  and X are in U (ii) the intersection
of two members of U is in U .(iii) the union of any number of members U is in U . The set X with U is
called a topological space the members uU  are called the open sets . let X be a topological space a subset
XN  with Nx  is called a neighborhood of x if there is an open set U with NUx  , for example
if X a metric space then the closed ball )( xD 
and the open ball )( xD 
are neighborhoods of x a subset C is
said to closed if CX is open
Definition 2.1.5
A function YXf : between two topological spaces is said to be continuous if for every open set U of Y
the pre-image )(
1
Uf

is open in X .

Operator’s Differentialgeometry With Riemannian Manifolds
Definition 2.1.6
Let X and Y be topological spaces we say that X and Y are homeomorphic if there exist continuous
function such that y
idgf  and X
idfg  we write YX  and say that f and g are homeomorphisms
between X and Y , by the definition a function YXf : is a homeomorphisms if and only if .(i) f is a
bijective .(ii) f is continuous (iii) 1
f is also continuous.
Definition 2.1.7 Coordinate Charts
A coordinate chart or just a chart on a topological n manifold M is a pair ),( U , Where U is an open subset
of M and UU
~
:  is a homeomorphism from U to an open subset n
RUU  )(
~
 .
Examples 2.1.8 Topological Manifolds] Spheres
Let n
S denote the (unit) n sphere, which is the set of unit vectors in 1n
R : }1:{
1


xRxS
nn
with the
subspace topology, n
S is a topological n manifold.
Definition 2.1.9 Projective spaces
The n dimensional real (complex) projective space, denoted by ))()( CPorRP nn
, is defined as the set of 1-
dimensional linear subspace of )
11  nn
CorR , )()( CPorRP nn
is a topological manifold.
Definition 2.1.10
For any positive integer n , the n torus is the product space )...(
11
SST
n
 .It is an n dimensional
topological manifold. (The 2-torus is usually called simply the torus).
Definition2.1.11 Boundary of a manifold
The boundary of a line segment is the two end points; the boundary of a disc is a circle. In general the boundary
of an n manifold is a manifold of dimension )1( n , we denote the boundary of a manifold M as M . The
boundary of boundary is always empty,  M
Lemma 2.1.12
(i)Every topological manifold has a countable basis of Compact coordinate balls. ( ii ) Every topological
manifold is locally compact.
Definitions 2.1.13 [ Transition Map]
Let M be a topological space n -manifold. If ),(),,(  VU are two charts such that  VU , the composite map.
(1) )()(:
1
VUVU 

 
Is called the transition map from  to .
Definition 2.1.14 [A smooth Atlas]
An atlas A is called a smooth atlas if any two charts in A are smoothly compatible with each other. A smooth
atlas A on a topological manifold M is maximal if it is not contained in any strictly larger smooth atlas. (This
just means that any chart that is smoothly compatible with every chart in A is already in A.
Definition 2.1.15 [ A smooth Structure ]
A smooth structure on a topological manifold M is maximal smooth atlas. (Smooth structure are also called
differentiable structure or 
C structure by some authors).
Definition 2.1.16 [ A smooth Manifold ]
A smooth manifold is a pair ,( M A), where M is a topological manifold and A is smooth structure on M . When
the smooth structure is understood, we omit mention of it and just say M is a smooth manifold.
Definition 2.1.17
Let M be a topological manifold: (i)Every smooth atlases for M is contained in a unique maximal smooth atlas.
(ii) Two smooth atlases for M determine the same maximal smooth atlas if and only if their union is smooth
atlas.
Definition 2.1.18
Every smooth manifold has a countable basis of pre-compact smooth coordinate balls. For example the General
Linear Group The general linear group ),( RnGL is the set of invertible nn  -matrices with real entries. It is a
smooth 2
n -dimensional manifold because it is an open subset of the 2
n - dimensional vector space ),( RnM ,
namely the set where the (continuous) determinant function is nonzero.
Definition 2.1.19[ Tangent Vectors on A manifold ]
Let M be a smooth manifold and let p be a point of M A linear map RMCX 

)(: is called a derivation at p if
it satisfies:
(2) XfpgXgpffgX )()()( 
Forall )(, MCgf

 . The set of all derivation of )( MC

at p is vector space called the tangent space to M at p ,
and is denoted by [ MT p
]. An element of MT p
is called a tangent vector at p .

Lemma 2.1.20 [ Properties of Tangent Vectors]
Let M be a smooth manifold, and suppose Mp  and MTX p
 . If f is a const and function, then 0Xf . If
(3) 0)()(  pgpf , then 0)( fpX .
Definition2.1.22 [Tangent Vectors to Smooth Curves ]
If  is a smooth curve (a continuous map MJ : ,where RJ  is an interval) in a smooth manifold M , we
define the tangent vector to  at Jt 
to be the vector .
(4) MT
dt
d
t tt )(
|)(

 
 





 
where 
t
dt
d | is the standard coordinate basis for RTt
. Other common notations for the tangent vector to  are





 
)(,)( 
t
dt
d
t

 and 





 
tt
dt
d
|

. This tangent vector acts on functions by :
(5)   )(
)(
||)( 



t
dt
fd
f
dt
d
f
dt
d
ft tt

 





 
Lemma 2.1.23
Let M be a smooth manifold and Mp  .Every  MTX p
 is the tangent vector to some smooth curve in M .
Definition 2.1.24 [ Lie Groups ]
A Lie group is a smooth manifold G that is also a group in the algebraic sense, with the property that the
multiplication map GGGm : and inversion map GGm : , given by 1
)(,),(

 ggihghgm , are both
smooth. If G is a smooth manifold with group structure such that the map GGG  given by 1
),(

 ghhg is
smooth, then G is a Lie group. Each of the following manifolds is a lie group with indicated group operation.
The general linear group ),( RnGL is the set of invertible nn  matrices with real entries. It is a group under
matrix multiplication, and it is an open sub-manifold of the vector space ),( RnM , multiplication is smooth
because the matrix entries of A and B . Inversion is smooth because Cramer’s rule expresses the entries of 1
A as
rational functions of the entries of A . The n torus )...(
11
SST
n
 is an n dimensional a Belgian group.
Definition 2.1.25 [ Generalized Tensor is Riemannian]
If an m-dimensional smooth manifold M is given a smooth every no degenerate symmetric covariant tensor
field of rank-2 , G then M is called a generalized tensor or metric tensor or metric of M . If G is positive
definite then M is called Riemannian manifold for a generalized Riemannian manifold ji
ji
dudugGM ,
specifies an inner product on the tangent space )( MT p
at every point Mp  for any )(, MTYX p
 .
(6)     ji
ij
YXpgYXGYX  ..
When G is positive definite, it is meaningful to define the length of a tangent vector and the angle between two
tangent vectors at the some point Ji
ij
XXgX  . Thus a Riemannian manifold is a differentiable manifold
which has a positive definite inner product on the tangent space at every point. The inner product is required to
smooth YX , are smooth tangent vector fields then YX , is a smooth on M
Definition 2.1.26 [ Smooth Parameterize Curve ]
ji
ji
dudugdS 
2
is independent of the choice of the local coordinate system i
u and usually called the metric form
or Riemannian metric )( dS is precisely the length of an infinitesimal tangent vector and is called the element of
are length . Suppose a  tuuC
ii
 and 10
ttt  is a continuous and piecewise smooth parameterized curve on
M ,then the are length of C is defined to be .
(7)








  dt
dtdt
dudu
gS
t
t
ji
ij
1
0
Remark 2.1.27
Exist a smooth is nonzero everywhere. The existence of a Riemannian metric on a smooth manifold is an
extraordinary result. In general there may not exist a non-positive. In the context of fiber bundles , the existence
of a Riemannian metric on M implies the existence of a positive definite smooth of bundle of symmetric
covariant tensor of order 2-on M, However for arbitrary vector bundles there may not exist a smooth which is
nonzero everywhere.
Theorem 2.1.28
Suppose M is an m-dimensional generalized Riemannian manifold then there exists a unique tensor – Free and
metric compatible connection on M , called the ( Levi-civet connecting of M ) Riemannian connection of M
Proof:

Suppose D is a torsion-free and metric – compatible connection on M , denote the connection matrix of D under
the local coordinates i
U by  j
i
WW  where kj
ik
j
i
duW  .Then we have ki
k
jkj
j
iij
gWgWdg  , and j
kj
j
ik
 Denote
that l
ilkik
j
kjij
j
ik
WgWg  , . Then its follows from that.
(8)  j
ikjjikk
ij
u
g



j
ikjjik
 is cycling the indices in we get j
ikjjikj
ik
u
g



and j
ikjjiki
ik
u
g



.And calculating we then obtain .
(9)





















k
jk
i
jk
j
ik
ji
u
g
u
g
u
g
2
1




















 l
ij
i
jl
j
ilkl
ij
k
u
g
u
g
u
g
g
2
1
The equation is ( Levi-civet connecting of M ) or ( Riemannian connection of M )
Definition 2.1.29 [ Smooth Curve in M ]
Let M be a Riemannian manifold and   M1,0: a smooth map i,e a smooth curve in M . The length of
curve is )(L and 








dcz
baz
ZF )( With dcba ,,, and 0 bcad , then 2
)(
)(
dcz
dz
bcadFdz

 and
.
(10)
















g
y
dydx
ybcad
dcz
dcz
dydx
bcadFg
2
22
22
2
2
2
2
)()(
)(
So these Movies transformation are isometrics of Riemannian metric on the upper half-plan.
2.3 : The Spectral Geometry of operators of Dirac and Laplace Type
We have also given in each a few additional references to relevant. The constraints of space have of necessity
forced us to omit many more important references that it was possible to include and we a apologize in a dance
for that. We a the following notational conventions , let ),( gM ( be compact Riemannian manifold of dim. M
with boundary M .Let Greek indices  , range from mto1 , and index a local system of coordinates
 m
xxx ...,,.........
1
 on the interior of M expand the metric in the form 

dxdxgdS 
2
were  
 xx
g  , and
where we adopt the Einstein convention of summing over repeated indices we let 
g be the inverse matrix the
Riemannian measure is given by  m
dxdxdx ...,,.........
1
 for  
gg det let  be the”levi-Civita” connection. We
expand 
 xxx
 
.,
. Where 
 are the Rm , are may then be given by.    YXXYYX
YXR ,
, 
And given by.
(11)    WZYXRgWZYXR ,),,(,,, 
We shall let Latin indices ji , range from 1 to m and index a local orthonormal frame  m
ee ...,,.........1
for the
components of the curvature tensor scalar curvature . Are then given by setting ikkijikkjiij
RandR   .
We shall often have an auxiliary vector bundle set V and an auxiliary given on V , we use this connection and
the “ Levi-Civita” connection to covariant differentiation , let dy be the measure of the induced metric on
boundary M , we choose a local orthonormal from near the boundary M , so that  m
e is the inward unit normal
. We let indices a,b range from 1 to m-1 and index the induced local frame  11
...,,......... m
ee for the tangent
bundle f the boundary , let  megba
eebL
a
,,
 denote the second fundamental form . we some over indices with the
implicit range indicated . Thus the geodesic curvature g
K is given by aag
LK  . We shall let denote multiple
tangential covariant differentiation with respect to the “Levi-Civita” connection the boundary the difference
between and being of course measured by the fundamental form.
Proposition 2.3.1[ Manifold admits a Riemannian Metric]
Any manifold a demits a Riemannian metric
Proof :
Take a converging by coordinate neighborhoods and a partition of unit subordinate to covering on each open set

U we have a metric 
i
i
dxg
2
 . In the local coordinates, define )( i
i
i
gg 
 this sum is well-defined

because the support of i
 . Are locally finite. Since 0i
 at each point every term in the sum is positive
definite or zero, but at least one is positive definite so that sum is positive definite.
Proposition 2.3.2[ The Geodesic Flow]
Consider any manifold M and its cotangent bundle )(
*
MT , with projection to the base MMTp )(:
*
, let
X be tangent vector to )(
*
MT at the point MTa
*
 then )()(
*
MTXD p
 so that ))(()( xDX pa
 
defines a conicala conical 1-form  on )(
*
MT in coordinates 
i
i
dyyyx ),( the projection p is
xyxp ),( so if
i
i
i
i
y
b
x
aX





  so if given take the exterior derivative ii
dydxdw  
which is the canonical 2-from on the cotangent bundle it is non-degenerate, so that the map )( wiX  from
the tangent bundle of )(
*
MT to its contingent bundle is isomorphism. Now suppose f is smooth function an
)(
*
MT its derivative is a 1-form df .Because of the isomorphism a above there is a unique vector field X on
)(
*
MT such that )( widf  from the g another function with vector field Y , then .
(12) g
wiwi
Y
XYXiXiiYdftY )()()( 
On a Riemannian manifold we shall see next there is natural function on )(
*
MT . In fact a metric defines an
inner on *
T as well as on T for the map ),(  XgX defines an isomorphism form T to *
T then
ki
j k
llkjji
gdxgdxgg 





  which means that ki
kj
gdxdxg ),(
*
where ki
g denotes the matrix to ki
g we
consider the function )(
*
MT defined by ),()(
*
aaa
gH   .
2.4 Maxima and Minima Lecture
Example 2.4.1
A Texas based company called (Hamilton’swares) sells baseball bats at a fixed price c . A field researcher has
calculated that the profit the company makes selling the bats at the price c is








 cccccp 1150
2
51
5
1
2000
1
)(
234
at what price should the company sell their bats to make the most money.
Intuitively what would we have to do solve this problem. We wish to know at what point c is this function )(cP
is maximized. we do not have many tools as moment to solve this problem so let’s try to graph the function and
guess at where the value should be.
Definition 2.4.2
Let f be function defined on an interval I containing c we say that f has an absolute maximum ( or a global
maximum ) value on I at c )()( cfxf  for all x contained in I . Similarly, we say that f has an absolute
minimum ( or a global minimum ) value on I at c if )()( cfxf  for all x contained in I . Those points together
are known as absolute global extreme.
Example 2.4.3
1)(
2
 xxf for   ,x remember this notation means for x living in the interval from negative infinity to
infinity . This can also be written as Rx  or in words as for all real x ,this function has an absolute minimum of
at the point 0x but no absolute maximum on the interval .
Example 2.4.4
1)(
2
 xxf for  2,2x remember closed brackets means we include the endpoints in our interval this function
has an absolute minimum of I at the point 0x and a absolute maximum of 51)2()2(
2
f at the points
2x and 2x .
Example 2.4.5
1)(
2
 xxf for  2,0x remember open brackets means we omit the endpoint in our interval .
Example 2.4.6
3
)( xxf  for   ,x , this function has no absolute minimum and no absolute maximum .
Definition 2.4.7[ Extreme value theorem]
A function have an a absolute maximum and minimum these examples seen to suggest that if we have a closed
interval then we’re in business.
Example 2.4.8
Consider the function .from the graph, it’s clear that this function has no absolute minimum or absolute
maximum but )( xf is defined on all of  2,0 the problem with this example is that the function is not
continuous.

(13)









214
2,1,05.1
10
)(
xifx
xif
xifx
xf
Theorem 2.4.9[Extreme value ]
Let )( xf be a continuous function defined on a close interval, then )( xf has an absolute maximum and an
absolute minimum on that interval .
[Notice]: that this says nothing about uniqueness. Remember the example 1)(
2
 xxf for  2,2x has two
points where the absolute maximum was obtained. Also note that functions that are not continuous and defined
on a closed interval can still have extreme.
Example 2.4.10
Consider the following function on  1,1 as function )( xf , this function is not continuous at 0 however it has a
global minimum of 0 of -3 because at all non-zero points this
function is sturdily positive.
(14)






03
0
)(
xif
xifx
xf
Definition 2.4.11
Let I be an open interval on which a function f is defined and suppose that Ic  .We say that c is a local
maximum value of f if )()( cfxf  for all x contained in some open interval of I .Similarly we say that c is a
local minimum value of f if )()( cfxf  for all x contained in some open interval I . These points together are
known as local extreme.
[Note] : Your textbook uses any arbitrary interval, but requires c to be an interior point.
[Note] : Global extreme of a function that occur on an open interval contained in our domain are also local
extreme.
Theorem 2.4.12 [ Fermat’s or local extreme ]
If a function )( xf has a local minimum or maximum at the point c and )(cf  exists, then 0)(  cf
Example 2.4.13
We look at xxf )( . Notice that this function is not differentiable 0x but since )0(0)( fxxf   we see
that it has a local minimum at 0 ( and in fact this is a global minimum ).
Definition 2.4.14
A critical point is a point c in the domain of f where 0)(  cf or )(cf  fails to exist . In fact all critical points
are candidates for extreme but it is not true that all critical points are extreme.
Example 2.4.15
Consider the function 3
)( xxf  .We saw before that this function has no maximum or minimum. However
2
3)( xxf  and 0)0(3)0(
2
f so the point 0x is a critical point of f that is not an extreme.
2.5 : [ Algorithm for finding global minima and maxima ]
Let f be a continuous function on a closed interval  ba , so that our algorithm satisfies the conditions the
conditions of the extreme value theorem:(i) Find all the critical points of  ba , , that is the points  bax , where
)( xf  is not defined or where 0)(  xf ( usually done by setting the numerator and denominator to zero ) call
these points n
xxx ,....,, 21
.(ii) Evaluate )(),(),(),.....,( 1
bfafxfxf n
that is evaluate the function at all the critical
points found from the previous step and the two end point values . (iii) The largest and the smallest values found
in the previous step are the global minimum and global maximum values.
Example 2.5.1
Compute the absolute maximum and minimum of  2,1243  onxx .
Solution
Our function is continuous ( and in fact differentiable ) everywhere . Hence we 46)(  xxf setting 0)(  xf
and solving yields xxxxf  3/26446)(0 . Now we evaluate f at 1,3/2 x and 2 ( that is the
critical points and the end points ) we get that .
(15)


























02)2(4)2(3)2(,92)1(4)1(3)1(,
3
2
4
3
2
4
3
2
3
3
2 22
2
fff
From this , we see that the absolute maximum is 9 obtained at 1x and the absolute minimum is  3/2 obtained
at  3/2x .

Example 2.5.2
Compute the critical points of 3/2
5)( xxf  .
Solution
We compute the derivative 3
1
3
10
)(

 xxf ,Now we check when the derivative is 0 and when it is undefined This
function is never 0 but happens to be undefined at 0 which is a point in our in domain . Hence the critical points
are just 0x .
2.5: The Geometric of Operators of Laplace and Dirac Type
In this subsection we shall establish basic definitions discuss operator of Laplace and of Dirac type introduce the
De-Ream complex and discuss the Boehner Laplacian and the weitzenbochformula. Let D be a second of
smooth sections  vC

of a vector bundle v over space M , expand  bxaxxaD v
v
 



where
coefficient  baa
v
,,

are smooth endomorphism’s of v, we suppress the fiber indices . We say that D is an
operator of Laplace type if 2
A ,on  vC

is said to be an operator of Dirac type if
2
A is an operator of Laplace
operator of Dirac type if and only if the endomorphism’s v
 satisfy the Clifford commutation relations
)(2 idg
vvv 
  . Let A be an operator of Dirac type and let v
v
dx  be a smooth 1-form on M we let
  v
v
v  define a Clifford module structure on V . This is independent of the particular coordinate system
chosen. We can always choose a fiber metric on so that  is skew adjoin. We can then construct a unitary
connection  on V so that 0  such that a connection is called compatible the endomorphism if  is
compatible we expand A
v
xv
A   
, A
 is tonsorial and does not depend on the particular coordinate system
chosen it does of course depend on the particular compatible connection chosen.
Definition 2.5.1 [ The De-Rahm Complex ]
The prototypical example is given by the exterior algebra, let  MC
p


be the space of smooth p forms. Let
   MCMCd
pp 1
:

 be exterior differentiation if 
is cotangent vector, Let wwext   :)( denote
exterior multiplication and let )(int  be the Dual , Interior multiplication, )int()()(   extv define module on
exterior algebra  M . Since   v
x
v
dxvd 
  . d is an operator of “ Direct type” the a associated Laplacian
  m
M
p
mmm
d  ...........
02
 decomposes as the “Direct sum” of operators of Laplace type p
m
 on the
space of smooth p forms  MC
p


on has  v
v
M
xggxg 
 

10
it is possible to write the p-form valued
Laplacian in an invariant form . Extend the “ Levi-Civita” conduction to act on tensors of all types .Let
vwg
v
M
w 

,
~
 define Buchner or reduced Laplacian , let R given the associated action of curvature tensor .
The “Weitzenbock” formula terms of the“Buchner Laplacian” in the form     

 RdxdxMM
2
1~

This formalism can be applied more generally.
Lemma 2.5.2[ Spinner Bundle]
Let D be an operator of Laplace type on a Riemannian manifold, there exists a unique connection  onV and
there exists a unique endomorphism E of V , so that  ED ii
 if we express D locally in the form
 bxaxxgD  



then the connection 1-form w of  and the endomorphism E are given by .
(16)    





 





 
wwwwxgbEandidgagw E
E
2
1
LetV be equipped with an auxiliary fiber metric, then D is self-adjoin if and only if  is unitary and E is self-
adjoin we note if D is the Spinner bundle and the “Lichnerowicz formula” with our sign convention that
)(
4
1
idJE  where J is the scalar curvature.
Definition 3.4.3 Heat Trace Asymptotic for closed manifold
Throughout this section we shall assume that D is an operator of Laplace type on a closed Riemannian
manifold ),( gM . We shall discuss the 2
L - spectral resolution if D is self-adjoin , define the heat equation
introduce the heat trace and the heat trace asymptotic present the leading terms in the heat trace . Asymptotic
references for the material of this section and other references will be cited as needed , we suppose that D is
self-adjoin there is then a complete spectral resolution of D on  vL
2
. This means that we can find a complete
orthonormal basis  n
 for  vL
2
where they n
 are a smooth sections toV which satisfy the equation nnn
D   .

2.6 : Inverse Spectral Problems in Riemannian geometry
In al-arguably one the simplest inverse problem in pure mathematics “ can on hear the shape of drum “
mathematically the question is formulated as follows , let  be a simply connected plane domain ( The
drumhead bounded by a smooth curve  ) , and consider the wave equation on  with . Dirichlet boundary
condition on  - ( the drumhead is clamped at boundary )
      intxUintxU
C
txu tt
0,,,
1
, 2

The function ),( txU is the displacement of drumhead as vibrates at position x at time t , looking for solutions
of the form    xvetxU
twi
Re,  (normal modes) leads to an eigenvalue problem for the Dirichlet Laplacian on 
Where 22
/ cw , we write the infinite sequins of Dirichlet eigenvalues for this problem as   

 1nn
 or
simply  

1nn
 , if the choice of domain  is clear in context , Kans question means the following is it possible
to distinguish “ drums “ 1
 and 2
 with distinct ( modulo isometrics ) bounding curves 1
 and 2
 simply by (
hearing ) all of the eigenvalues of Dirichlet Laplacian some surprising and interesting results are obtained by
considering the heat equation on  with
Dirichlet boundary conditions, which given rise to the same boundary value
problem as before the heat equation is :
(17)
   
 
   







xfxU
ontxU
intxUtxU t
0,
0,
,,

Where  txU , is the temperature at point x and time t , and f(x) is the initial temperature distribution. This
evolution equation is formal solution.      xfetxU
t 
, . Where the operator t
e can be calculated using the
spectral resolution of  . Indeed if  xj
 is the normalized Eigen function of the boundary value problem with
eigenvalue j
 the operator t
e has integral Kernel ),,( yxtk the heat Kernelgiven by .
(18)    yxeyxtk
j
t







1
),,(
The trace of ),,( yxtk is actually a spectral in variant by ( we can compute ).





1
),,(
j
t j
eyxtk

[Not] that the function determines the spectrum  

1nn
 , to analyze the geometric content of spectrum, one
calculates the by completely different method one constructs the heat kernel by perturbation from the explicit
heat kernel for the plane, and then on computes the trace explicitly. It turns out that the trace has a small-t
asymptotic expansion.
(19)    ..........
4
1
,, 1210


taaa
t
dxtxxk

Where    lengthaareaa  10
, , Al though a strict derivation is a bit involved which shows why 0
a and 1
a
should given the area of  and length of  the heat kernel in the plan is .    tyx
t
tyxk 

4/exp
4
1
,,
2
0
 , we
expect particle that for small times    txxktxxK ,,,, 0
 ( a Brownian particle starting out the interior doesn’t the
boundary for a time of order t ).
(20)       

are
t
dxtxxkdxtxxK
4
1
,,,, 0
For times of order t , boundary effects become important we can approximate the heat kennel near the
boundary locally by ( method images ) locally the boundary looks the line 211
0 xxtheinx  plane , letting
*
xx  be ,      
*
02
,,,,,, tyxktxxktxxK  vanishes 01
x hence  
t
e
txxK
t


4
1
,,
/2
2


 where  . Is the distance
from x to the boundary , writing the volume integral for the additional term as an integral over the boundary
curve and distance from the boundary dsde
t
t




/2
0 4
1 

  we have.
(21)  
   











 tt
length
t
area
dxtxxK
1
2
1
44
,,



It follows that the is spectral set of a given (drum )  contains only drums with the same area and perimeter here
we will briefly discuss the generalization of kais problem and some of the known results. A Riemannian

manifold of dimension n is a smooth n-dimensional manifold M . Equipped with a Riemannian metric g which
defines the length of tangent vectors and determines distances and angles on the manifold . The metric also
determines the Riemann curvature tensor of M . In two dimensions , the Riemannian curvature tensor is in turn
determined by the scalar curvature, and in three dimensions it is completely determined by the Ricci curvature
tensor. If M is compact the associated Laplacian has infinite set of discrete eigenvalues   1

nn
 what is the
geometric content of the spectrum for a compact Riemannian manifold. Constructs a pair 16-dimensional tori
with with the same spectrum. The torii nn
TandT 21
are quotients of n
R by lattices 21
 and of translations of n
R .
Since the tow tori are isometric of and only if their lattices are congruent, it suffices to construct a pair of non-
congruent 16-dimensional lattices whose a associated tori have the same spectrum .To understand the analysis
involved in Milnor’s construction consider the following simple “ trace formula” for a torus  /
nn
RT which
computes the trace of the heat kernel on a torus in terms of lengths of the lattice vectors to the heat kernel on the
torus is given by the formula.    tywxktxxK
w
,,,, 0
 

  
 




w
tw
n
e
t
Tvol
dxtxxK
4/
0
2
4
,,

Milnor noted that there exist non-congruent lattices in 1-dim. With the same set of “ length “  ww : first
discovered by the trace of the heat kernel determines the spectrum and the heat trace is in turn determined by the
lengths, it follows that the corresponding non-isometric tori have the same spectrum.
Example 2.6.1 [ Riemannian Manifold with Same Spectrum ]
Riemannian manifold with the same spectrum letter constructed continuous families of is spectral manifold in
sufficiently high dimension 5n two major questions remained:
(i) can one show that the is spectral set of given manifold at finite in low dimension .
(ii) can on find counterexamples for Kicks original problem , can one construct is spectral , non-congruent
planar
Definition 2.6.2 [ Some Positive Results ]
In proved one of the first major positive results on is spectral sets of surfaces and planar domains informally. A
sequence of planar domains j
 converges in 
C since to a limiting non-degenerate set compact surfaces j
S
converges in 
C sense to limiting non-degenerate surface S , converge in 
C sense to a positive definite metric on
S
Theorem 2.6.3
(i) Let j
 be a sequence of is spectral planar domains there is a subsequence which converges in 
C sense to no
degenerate limiting surface.
(ii) Let j
S be a sequence of is spectral compact surfaces there is a subsequence of the j
S converging in 
C sense
to a non-degenerate surface S .
Theorem 2.6.4
Suppose  j
M is a sequence if is spectral manifold such that either : (i) All of the j
M - have negative sectional
curvatures .(ii) All of the j
M -have Ricci curvatures bounded below .Then there are finitely many
diffoemorphism types and there is a subsequence which convergent in 
C to a nodegenerate limiting manifold .
(22) 







 00
1
,....,)( t
dt
d
t
dt
d
t
dt
d
n

we many k bout smooth curves that is curves with all continuous higher derivatives cons the level surface
  cxxxf
n
,...,,
21
of a differentiable function f where
i
x to  thi  coordinate the gradient vector of f at
point )(),....,(),(
21
PxPxPxP
n
 is 









 n
x
f
x
f
f ,.....,1
is given a vector ),...,(
1 n
uuu  the direction
derivative n
nu
u
x
f
u
x
f
uffD





 ...
1
1
, the point P on level surface  n
xxxf ,...,,
21
the tangent is
given by equation. 0)()()(....))(()(
11
1






PxxP
x
f
PxxP
x
f nn
n
For the geometric views the tangent space shout consist of all tangent to smooth curves the point P , assume
that is curve through 0
tt  is the level surface.   cxxxf
n
,...,,
21
,   ctttf
n
)(),....,(),(
21
 by taking
derivatives on both   0))()(....)(( 01






tP
x
f
tP
x
f n
n
 and so the tangent line of  is really normal
orthogonal to f , where  runs over all possible curves on the level surface through the point P . The surface

M be a 
C manifold of dimension n with 1k the most intuitive to define tangent vectors is to use curves ,
Mp  be any point on M and let   M  ,: be a 1
C curve passing through p that is with pM )(
unfortunately if M is not embedded in any N
R the derivative )( M  does not make sense ,however for any
chart  ,U at p the map    at a 1
C curve in n
R and tangent vector   )( Mvv  is will defined the
trouble is that different curves the same v given a smooth mapping MNf : we can define how tangent
vectors in NT p
are mapped to tangent vectors in MTq
with  ,U choose charts )( pfq  for Np  and
 ,V for Mq  we define the tangent map or flash-forward of f as a given tangent vector.
(23)   NTX pp
  and      ffMTfd p
**
,:
A tangent vector at a point p in a manifold M is a derivation at p , just as for n
R the tangent at point p form
a vector space )( MT p
called the tangent space of M at p , we also write )( MT p
a differential of map
MNf : be a 
C map between two manifolds at each point Np  the map F induce a linear map of
tangent space called its differential p , NTNTF pFp )(*
:  as follows it NTX pp
 then )(* p
XF is the
tangent vector in MT pF )(
defined .
(24)     )(,)(*
MCfRFfXfXF pp

 
The tangent vectors given any 
C - manifold M of dimension n with 1k for any Mp  ,tangent vector to
M at p is any equivalence class of 1
C - curves through p on M modulo the equivalence relation defined in
the set of all tangent vectors at p is denoted by MT p
we will show that MT p
is a vector space of dimension n of
M . The tangent space MT p
is defined as the vector space spanned by the tangents at p to all curves passing
through point p in the manifold M , and the cotangent MT p
*
of a manifold at Mp  is defined as the dual
vector space to the tangent space MT p
, we take the basis vectors 







 ii
x
E for MT p
and we write the basis
vectors MT p
*
as the differential line elements ii
dxe  thus the inner product is given by.
(25) j
i
i
dxx  ,/
Theorem 2.6.5[Bounded Harmonic Function ]
Suppose that  is a bounded, connected open set in n
R and     CCU
2
is harmonic in  then.
UU 
 maxmax and UU 
 minmin
Proof :
Since U is continuous and  is compact , U attain its global maximum and minimum on  , ifU attains a
maximum or minimum value at interior point then U is constant by otherwise both extreme values are attained
in the boundary .In either cases the result follows let given a second of this theorem that does not depend on
the mean value property .Instated we us argument based on the non-positivity of the second derivative at an
interior maximum . In the proof we need to account for the possibility of degenerate maxima where the second
derivative in zero . For    
2
,0 xxUxUlet 

 . Then 02  

nU ,since U is harmonic .if 
U attained
a local maximum at an interior point then 0

U by the second derivative test thus 
U no interior maximum,
and it attains its maximum on the boundary .If,  xallforRx , , if follows that.
(26) 2
RUSupUSupUSupUSup 

 
letting 
 0 ,we get that USupUSup 
 .An application for the same a grummet to u given in,
UU 
 infinf .and the result follows . Sub harmonicfunction satisfy a maximum principle UU 
 minmin
,while sub harmonic function satisfy a minimum principle UUU 
 minmin for all x . Physical terms,
this means for example that the interior ofabounded region which contains no heat sources on heat sources or
sinks cannot be hotter that the maximum temperature on the boundary or colder than the minimum temperature
on the boundary .The maximum principle given a uniqueness result for ( Dirichlet problem) for the poison
equation .
Definition 2.6.6
Let 1
M and 2
M be differentiable manifolds a mapping 21
: MM  is a differentiable if it is differentiable
objective and its inverse 1
 is diffoemorphism if it is differentiable  is said to be a local diffoemorphism at
Mp  if there exist neighborhoods U of p and V of )( p such that VU : is a diffoemorphism , the

notion of diffoemorphism is the natural idea of equivalence between differentiable manifolds , its an immediate
consequence of the chain rule that if 21
: MM  is a diffoemorphism then.
(27) 2)(1
: MTMTd pp 
 
Is an isomorphism for all 21
: MM  in particular , the dimensions of 1
M and 2
M are equal a local
converse to this fact is the following 2)(1
: MTMTd pp 
  is an isomorphism then  is a local
diffoemorphism at p from an immediate application of inverse function in n
R , for example be given a
manifold structure again A mapping NMf 

:
1
in this case the manifolds N and M are said to be
homeomorphism , using charts ),( U and ),( V for N and M respectively we can give a coordinate
expression NMf :
~
Theorem 2.6.7 [ Compact Riemannian Manifolds ]
Let M be compact Riemannian Manifold with or without boundary    .,, RFCMfM  proper function
satisfying.       srforXprxFXprxFsr  ,,,,,, There exist a function    .,0,0: W satisfying
  00  twhentW and   .00 W
Definition 2.6.8
Let 1
1

M and 1
2

M be differentiable manifolds and let 21
: MM  be differentiable mapping for every
1
Mp  and for each 1
MTv p
 choose a differentiable curve 1
),(: M  with pM )( and
v )0( take   the mapping 2
)(: MpTd p 
 by given by )()( Mvd   is line of  and
1
2
1
1
:

 MM be a differentiable mapping and at 1
Mp  be such 21
: MTMTd p 
  is an isomorphism
then  is a local homeomorphism .
Proposition2.6.9
Let n
M 1
and m
M 1
be differentiable manifolds and let 21
: MM  be a differentiable mapping , for every
1
Mp  and for each 1
MTv p
 choose a differentiable curve 1
),(: M  with po )( , vo  )(
take   the mapping 2)(1
: MTMTd pp 
  given by )()( ovd p
  is a linear mapping that dose
not depend on the choice of  .
Theorem 2.6.10
The tangent bundle TM has a canonical differentiable structure making it into a smooth 2n-dimensional
manifold, where N=dim. The charts identify any )()( TMMTUU pp
 for an coordinate neighborhood
MU  , with n
RU  that is Hausdorff and second countable is called ( The manifold of tangent vectors )
Definition 2.6.11
A smooth vectors fields on manifolds M is map TMMX : such that:(i) MTPX p
)( for every G .(ii) in
every chart X is expressed as )/( ii
xa  with coefficients )( xa i
smooth functions of the local coordinates i
x .
Theorem2.6.12
Suppose that on a smooth manifold M of dimension n there exist n vector fields  )()2()1(
....,,,
n
xxx for a basis
of MT p
at every point p of M , then MT p
is isomorphic to n
RM  m here isomorphic means that TM and
n
RM  are homeomorphism as smooth manifolds and for every Mp  , the homeomorphism restricts to
between the tangent space MT p
and vector space   n
i
RP  .
Proof:
define TMMTa p


: on other hand , for any n
RM  for some Ra i
 now define
 n
n
RMaasTMa  ,....,:)(: 1


is it clear from the construction and the hypotheses of theorem that
 and 1
 are smooth using an arbitrary chart n
RMU : and corresponding chart.
(28) mn
RRTMUT 

)(:
1

Definition 2.6.13 [ Direct Computationof The Spectrum]
The first of those is straightforward: direct computation it rarely possible to explicitly compute the spectrum of
a manifold were actually discovered via this method . Milnor’s example mentioned above consists of two is
spectral factory-quotients of Euclidean space by lattices of full rank being one of full rank being one of the few
examples of Riemannian manifolds whose spectra can be computed explicitly spherical space forms –
quotients of spheres by finite groups of orthogonal transformations acting without fixed points form another
class of examples of manifolds is spectral for the Laplacian acting on p-forms for kp  but not for the

Laplacian acting on p-forms for 1 kp (recall that a lens space is spherical space form where the group is
cyclic .Definition
3.5.14 Tensors on A vector Space
A tensor  on V is by definition a multiline map .
(29)






 RVVVV
sr
**
..........: 

V denoting the dual space toV , r its covariant order, and s its contra variant order.(Assume 0r or 0s ).
Thus  assigns to each r-tuple of elements of V and s-tupelo of elements of *
V a real number and if for each k,
srk 1 , we hold every variable except the k-th fixed, then  Satisfies the linearity condition .
(30)      1111
,...,...,,...,...,,...,..., kkkk
vvvvvvv   .
For all R , and Vvv kk
, or  
V .respectively For a fixed (r,s) we let )(V
r
s
be the collection of all tensors on
V of covariant order r and contra variant order s. We know that as a function from **
....... VVVV  to R
they may be added and multiplied by scalars elements of R. With this addition and scalar Multiplication r
s
V )( is
a vector space, so that if 21
,  )(V
r
s
and ,, 21
R then 2211
  , defined in the way alluded to above, that
is, by.
(31)
   
   




,...,,...,
,...,
21222111
212211
vvvv
vv


is multiline, and therefore is in r
s
V )( . Thus r
s
V )( has a natural vector space structure.
Theorem 2.6.15
With the natural definitions of addition and multiplication by elements of R the set r
s
V )( of all tensors of order
),( sr on V forms a vector space of dimension sr
n

.
Definition 2.6.16 [ Tensor Fields]
A 
C covariant tensor field of order r on a 
C - manifold M is a function  which assigns to each MP 
an element P
 of   
r
P
MT and which has the additional property that given any 

C Vector fields on an open
subset U of M , then  r
XX ,...,1
 is a 
C function on U , defined by ,      rPPPr
XXPXX ,....,,..., 11
  . We
denote by  
r
M the set of all 

C covariant tensor fields of order r on M .
Definition 2.6.17
We shall say that r
V , r
V a vector space, is symmetric if for each rji  ,1 , we have :
  ),...,,...,,...,(,...,,...,,..., 11
vvvvvvvv ijrrij
  .Similarly, if interchanging the (i-th) and (j-th) variables, rji  ,1
Changes the sign.
(32)   ),...,,...,,...,(,...,,...,,..., 11
vvvvvvvv ijrrij
 
then we say  is skew or anti symmetric or alternating; covariant tensors are often called exterior forms. A
tensor field is symmetric (respectively, alternating) if it has this property at each point.
2.7: Geometrid Maximum and principle Riemannian manifolds
The version of the analytic principle given by:
(i) 0
U is lower semi – continuous and   00
HUM  in the sense of support function.(ii) 1
U is upper – semi –
continuous and   00
HUM  in the sense function with a one – sided Hessian bound .(iii) 01
UU  in  and
01
UU  is locally a 1,1
C - function in  finally if ij
a and b are locally ,2k
C function in  . In particular if ij
a and
b are smooth is 01
UU  , n
R is specially natural in Lorentz Ian setting as 0
C space like hyper surfaces in
definition   rrpdpS r
 ),(exp,:,
 them r
S , contains   and neighborhood of   is smooth , at  
pointing unit normal 0r and  MTk  can a lows be locally represented as a graphs also applies to hyper
surfaces in Riemannian manifolds that can be represented locally as graphs. We first state our conventions on
the sign of the second fundamental form and the mean curvature to fix choice of signs a Lorentz Ian manifold
 gM . .
Definition 2.7.1[ Space time and Space like]
A subset MN  of that space-time  gM . is 0
C space likehyper surface , if for each Np  , there is a
neighborhood U of p in M so that VN  is causal and edge less in U .

Remark 2.7.2
In This definition not that if  UUND , is the domain of dependence of inU , then  UUND , is open in M and
UU  is a Cauchy hyper surface is globally hyperbolic thus by replacing U by  UUND , we can assume the
neighborhood U is the last definition is globally hyperbolic and that UU  in a Cauchy surface in U . In
particular a 0
C space like hyper surface is a topological.Let  gM . be a space-time and let 0
N and 1
N be two 0
C
space like hyper surfaces in  gM . which meet at a point q . Say that no is locally to the future of 1
N near q if for
some neighborhood U of P in which 1
N is a causal and edgeless  

,10
NJUN where  

,1
NJ is causal
future of 1
N in U .
Definition 2.7.3 [Multiplication of Tensors on Vector Space]
Let V be a vector space and V are tensors. The product of  and  , denoted
  is a tensor of order sr  defined by ),....,(),....,(),....,...,...( 1111 srrrsrrr
vvvvvvvv 
  .
The right hand side is the product of the values of  and  .The product defines a mapping    , of x
 V
r   V
sr 
.
Theorem 2.7.4
The product  V
r
o  V
r
  V
sr 
just defined is bilinear and associative. If n
 ,....,
1
is a basis of .
(33) 

V
 
!!....!
!...
....
21
21
21
k
k
k
rrr
rrr 
 
then     nii r
ii r
 ,...,1/.... 1
1
 is a basisof   r
V .Finally VWF :*
is linear, then
     
***
FFF  .
Proof:
Each statement is proved by straightforward computation. To say that  is bilinear means that if  , are
numbers and   V
r
, then      .2121
  Similarly for the second variable. This is
checked by evaluating each Side on sr  vectors of V ; in fact basis vectors suffice because of linearity
Associatively,      ,is similarly verified the products on both sides being defined in the
natural way. This allows us to drop the parentheses. To see that r
ii
  .....1
from a basis it is sufficient to note
that if n
ee ,...,1
is the basis of V dual to n
 ,....,
1
, then the tensor r
ii ,...,1
 previously defined is exactly r
ii
  .....1
.This follows from the two definitions:
(36)  
   
    








rr
rr
jj
ii
jjiiif
jjiiif
ee
r
r
,...,,...,1
,...,,...,0
,...,
11
11...
1
1
          r
rr
r
r
r
i
j
i
j
i
jj
i
j
i
j
i
jj
ii
eeeee  ...,...,,...,.... 2
2
1
12
2
1
1
1
1
 ,which show that both tensors have the same values
on any (ordered) set of r basis vectors and are thus equal. Finally, given ,:*
VWF  if Www sr

,..,1
, then
(35)      ))(),...,((,..., *11
*
srsr
wFwFwwF 
  =   ))(),....,(()(),...,( 1*1* srrr
wFwFwFwF 
 =
    ).,.....,( 1
**
sr
wwFF 
 
Which proves      
***
FFF  and completes the proof.
Theorem 2.7.8 [Multiplication of Tensor Field on Manifold]
Let the mapping   Srsr
MMM

 )()( just defined is bilinear and associative. If  n
 ,....,
1
is a basis of  
1
M ,
then every element  
r
M is a linear combination with 
C coefficients of    },...,1/....{ 1
1
nii r
ii r
  . If
MNF : is a 
C mapping, M and   M
s
, then      
***
FFF  , tensor field on .N
Proof:
Since two tensor fields are equal if and only if they are equal at each point, it is only necessary to see that these
equations hold at each point, which follows at once from the definitions and the preceding theorem .
Corollary 2.7.9
Each r
U including the restriction toU of any covariant tensor field on M , has a unique expression form
 rr
r
r
ii
i i
ii
a     ........
1
1
....
. Where at each point  rr
iiii
EEaU ,...,,
11
....
 are the Components of  in the basis
 r
ii
  .....1
and is 
C function on .U
2.8 : Tangent Space and Cotangent Space
The tangent space )(MT p
is defined as the vector space spanned by the tangents at p to all curves passing
through point p in the manifold M . And The cotangent space )( MT p

of a manifold, at Mp  is defined as the

dual vector space to the tangent space )(MT p
. We take the basis vectors ii
x
E

 for )(MT p
, and we write the
basis vectors for )( MT p

as the differential line elements ii
dxe  Thus the inner product is given by
j
i
i
i dx
x


 , .
Definition 2.8.1 [ Wedge Product ]
Carton’s wedge product, also known as the exterior Product, as the ant symmetric tensor product of cotangent
space basis elements.
(36) )(
2
1
dxdydydxdydx   dxdy 
Note that, by definition, 0 dxdx . The differential line elements dx and dy are called differential 1-forms or
1-form; thus the wedge product is a rule for construction g 2-forms out of pairs of 1-forms.
Definition 2.8.2
Display equations should be broken and aligned for two-column display unless spanning across two columns is
essential. Equations should be centered with equation numbers set flush right. If using Math Type, use the .
Definition 2.8.3 [ Vector Analysis one Method Lengths]
Classical vector analysis describes one method of measuring lengths of smooth paths in 3
R if   3
1,0: Rv  is
such a paths, then its length is given by length dttvv )( . Where  v is the Euclidean length of the tangent
vector )(t , we want to do the same thing on an abstract manifold ,and we are clearly faced with one problem ,
how do we make sense of the length )(tv ,obviously , this problem can be solved if we assume that there is a
procedure of measuring lengths of tangent vectors at any point on our manifold The simplest way to do achieve
this is to assume that each tangent space is endowed with an inner product ( which can vary point in a smooth ).
Definition 2.8.3
A Riemannian manifold is a pair ).( gM consisting of a smooth manifold M and a metric g on the tangent
bundle ,i.ea smooth symmetric positive definite tensor field on M . The tensor g is called a Riemannian metric
on M . Two Riemannian manifold are said to be isometric if there exists a diffoemorphism 21
: MM 
such that 21
: gg 

 If ).( gM is a Riemannian manifold then, for any Mx  the restriction
RMTMTg xxx
 )()(: 21
. Is an inner product on the tangent space )( MT x
we will frequently use thee
alternative notation ),(),(  xx
g the length of a tangent vector )( MTv x
 is defined as usual
 
2/1
, vvgv xx
 . If   Mbav ,: is a piecewise smooth path, then we defined is length by

b
a
dttvvL )()( . If we choose local coordinates ),....,(
1 n
xx on M ,then we get a local description of g as.
(37)   





















ji
ji
ji
ji
xx
ggdxdxgg ,,,
Proposition 2.8.4
Let be a smooth manifold, and denote by M
R the set of Riemannian metrics on M ,then M
R is a non –empty
convex cone in the linear of symmetric tensor
Proof :
The only thing that is not obvious is that M
R is non-empty we will use again partitions of unity . Cover M by
coordinate neighborhoods A
U 
)( . Let j
x be a collection of local coordinates on 
U . Using these local
coordinates we can construct by hand the metric 
g
on 
U by    
n
dxdxg  ...
1
now , pick a partition of unity )(0
MCB

 subordinated to cover 
U (i.e) there exists a map AB : such
that B
UB 
 
 then define 


B
Bgg

 )( The reader can check easily g is well defined ,and it is
indeed a Riemann metric on M .
Example 2.8.5 [ The Euclidean Space]
The space n
R has a natural Riemann metric n
dxdxg ,....,
1
0
 The geometry of  0
, gR
n
is the classical
Euclidean geometry
Example 2.8.6 [ Induced Metrics On Sub manifolds ]
Let  gM , be Riemann manifold and MS  a sub manifold if MS  , denotes the natural inclusion then
we obtain by pull back a metric on SggigS
S
/, 

,. For example , any invertible symmetric nn  matrix

defines a quadratic hyper surface in n
R by  1),(,  xARxH x
n
A
where   , denotes the Euclidean
inner on n
R , A
H has a natural .
Remark 2.8.7
On any manifold there exist many Riemannian metrics , and there is not natural way of selecting on of them .
One can visualize a Riemannian structure as defining “ shape ” of the manifold . For example , the unit sphere
1
222
 zyx , is diffeomorphic to the ellipsoid       13/2/1/
32222
 zyx ,but they look “different” by
However , appearances may be deceiving in is illustrated the deformation of a cylinder they look different ,but
the metric structures are the same since we have not change length of curves on our sheep . the conclusion to be
drawn from these two examples is that we have to be very careful when we use the attribute “different”.
Example 2.8.8[ The Hyperbolic Plane ]
The Poincare model of the hyperbolic plane is the Riemannian manifold  gD , where D is the unit open disk
in the plan n
R and the metric g is given by .
(39)  








22
22
1
1
dydx
yx
g
Example 2.8.9
Left Invariant Metrics on lie groups Consider a lie group G ,and denote by G
L its lie algebra then any inner
product  , on G
L induces a Riemannian metric g
h  , on G defined by.
(40)







)(,:
)(,,),(
11
GTyXGg
YLXLyxyxh
g
gggg
Where )()(:)( 1
1
GTGTL gg


is the differential at Gg  of the left translation map 1
g
L . One checks easily
that check easily that the correspondence  ,gG is a smooth tensor field, and it is left invariant (i,e)
GghhL g


. If G is also compact, we can use the averaging technician to produce metrics which are
both left and right invariant.
III. Conclusion
The paper study Riemannian differenterentiable manifolds is a generalization of locally Euclidean n
E in every
point has a neighbored is called a chart homeomorphism, so that many concepts from as differentiability
manifolds. We give the basic definitions, theorems and properties of Laplacian Riemannian manifolds becomes
the spectrum of compact support M and Direct commutation of the spectrum, and spectral geometry of operators
de Rahm.
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First Author
Dr. : Mohamed Mahmoud Osman- (phd)
Studentate the University of Al-Baha –Kingdom of Saudi Arabia
Al-Baha P.O.Box (1988) – Tel.Fax : 00966-7-7274111
Department of mathematics faculty of science
[1]
E-mail: mohm.mohm.osm@gamil.com
[2]
E-mail: moh_moh_os@yahoo.com
Tel. 00966535126844

Operator’s Differential geometry with Riemannian Manifolds

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