Open access peer-reviewed chapter

# Spectral Theory of Operators on Manifolds

Written By

Paul Bracken

Submitted: 28 April 2016 Reviewed: 29 November 2016 Published: 18 January 2017

DOI: 10.5772/67095

From the Edited Volume

## Manifolds

Edited by Paul Bracken

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## Abstract

Differential operators that are defined on a differentiable manifold can be used to study various properties of manifolds. The spectrum and eigenfunctions play a very significant role in this process. The objective of this chapter is to develop the heat equation method and to describe how it can be used to prove the Hodge Theorem. The Minakshisundaram‐Pleijel parametrix and asymptotic expansion are then derived. The heat equation asymptotics can be used to give a development of the Gauss‐Bonnet theorem for two‐dimensional manifolds.

### Keywords

• manifold
• operator
• differential form
• Hodge theory
• eigenvalue
• partial differential operator
• Gauss‐Bonnet

## 1. Introduction

Topological and geometric properties of a manifold can be characterized and further studied by means of differential operators, which can be introduced on the manifold. The only natural differential operator on a manifold is the exterior derivative operator which takes k ‐forms to k + 1 forms. This operation is defined purely in terms of the smooth structure of the manifold, used to define de Rham cohomology groups. These groups can be related to other topological quantities such as the Euler characteristic. When a Riemannian metric is defined on the manifold, a set of differential operators can be introduced. The Laplacian on k ‐forms is perhaps the most well known, as well as other elliptic operators.

On a compact manifold, the spectrum of the Laplacian on k ‐forms contains topological as well as geometric information about the manifold. The Hodge theorem relates the dimension of the kernel of the Laplacian to the k ‐th Betti number requiring them to be equal. The Laplacian determines the Euler characteristic of the manifold. A sophisticated approach to obtaining information related to the manifold is to consider the heat equation on k ‐forms with its solution given by the heat semigroup [13].

The heat kernel is one of the more important objects in such diverse areas as global analysis, spectral geometry, differential geometry, as well as in mathematical physics in general. As an example from physics, the main objects that are investigated in quantum field theory are described by Green functions of self‐adjoint, elliptic partial differential operators on manifolds as well as their spectral invariants, such as functional determinants. In spectral geometry, there is interest in the relation of the spectrum of natural elliptic partial differential operators with respect to the geometry of the manifold [46].

Currently, there is great interest in the study of nontrivial links between the spectral invariants and nonlinear, completely integrable evolutionary systems, such as the Korteweg‐de Vries hierarchy. In many interesting situations, these systems are actually infinite‐dimensional Hamiltonian systems. The spectral invariants of a linear elliptic partial differential operator are nothing but the integrals of motion of the system. There are many other applications to physics such as to gauge theories and gravity [7].

In general, the existence of nonisometric isospectral manifolds implies that the spectrum alone does not determine the geometry entirely. It is also important to study more general invariants of partial differential operators that are not spectral invariants. This means that they depend not only on the eigenvalues but also on the eigenfunctions of the operator. Therefore, they contain much more information with respect to the underlying geometry of the manifold.

The spectrum of a differential operator is not only studied directly, but the related spectral functions such as the spectral traces of functions of the operator, such as the zeta function and the heat trace, are relevant as well [8, 9]. Often the spectrum is not known exactly, which is why different asymptotic regimes are investigated [10, 11]. The small parameter asymptotic expansion of the heat trace yields information concerning the asymptotic properties of the spectrum. The trace of the heat semigroup as the parameter approaches zero is controlled by an infinite sequence of geometric quantities, such as the volume of the manifold and the integral of the scalar curvature of the manifold. The large parameter behavior of the traces of the heat kernels is parameter independent and in fact equals the Euler characteristic of the manifold. The small parameter behavior is given by an integral of a complicated curvature‐dependent expression. It is quite remarkable that when the dimension of the manifold equals two, the equality of the short‐ and long‐term behaviors of the heat flow implies the classic Gauss‐Bonnet theorem. The main objectives of the chapter are to develop the heat equation approach with Schrödinger operator on a vector bundle and outline how it leads to the Hodge theorem [12, 13]. The heat equation asymptotics will be developed [14, 15] andit is seen that the Gauss‐Bonnet theorem can be proved for a two‐dimensional manifold based on it. Moreover, this kind of approach implies that there is a generalization of the Gauss‐Bonnet theorem as well in higher dimensions greater than two [16, 17].

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## 2. Geometrical preliminaries

For an n ‐dimensional Riemannian manifold M , an orthonormal moving frame { e 1 , , e n } can be chosen with { ω 1 , , ω n } the accompanying dual coframe which satisfy

ω i ( e j ) = δ i j , i , j = 1, , n E1

It is then possible to define a system of one‐forms ω i j and two‐forms Ω i j by solving the equations,

X e i = j ω j i ( X ) e j , R ( X , Y ) e i = j Ω j i ( X , Y ) e j E2

It then follows that the Christoffel coefficients and components of the Riemann tensor for M are

ω j i ( e k ) = a ω a j ( e k ) e a , e i g = e k e j , e i g = Γ k j i E3
Ω i j ( e k , e s ) = a Ω a j ( e k , e s ) e a , e i g = R ( e k , e s ) e j , e i g = R k s j i E4

The inner product induced by the Riemannian metric on M is denoted here by , : Γ ( T M ) × Γ ( T M ) F ( M ) and it induces a metric on Λ k ( M ) as well. Using the Riemannian metric and the measure on M , an inner product denoted , : Λ k ( M ) × Λ k ( M ) R can be defined on Λ k ( M ) so that for α , β Λ k ( M ) ,

α , β = M α , β g d v M E5

where if ( x 1 , , x m ) is a system of local coordinates,

d v M = det ( g i j ) d x 1 d x m

is the Riemannian measure on M . Clearly, α , β is linear with respect to α , β and α , α 0 with equality if and only if α = 0 . Hodge introduced a star homomorphism * : Λ k ( M ) Λ n k ( M ) , which is defined next.

Definition 2.1. (i) For ω = i 1 < < i k f i 1 i k ω i 1 ω i k , define

* ω = i 1 < < i k j 1 < < j n k f i 1 i k ò ( i 1 , , i k , j 1 , , j n k ) ω j 1 ω j n k ,

where ò is 1 , 1 , or 0 depending on whether ( i 1 , , i k , j 1 , , j n k ) is an even or odd permutation of ( 1, , n ) , respectively.

(ii) If M is an oriented Riemannian manifold with dimension n , define the operator

δ = ( 1 ) n k + n + 1 * d * : Λ k ( M ) Λ k 1 ( M ) E6

In terms of the two operators d and δ , the Laplacian acting on k ‐forms can be defined on the two subspaces

Λ even ( M ) = even Λ k ( M ) , Λ odd ( M ) = odd Λ k ( M ) E7

The operator d + δ can be regarded as the operators on these subspaces,

D 0 = d + δ : Λ even ( M ) Λ odd ( M ) , D 1 = d + δ : Λ odd ( M ) Λ even ( M ) E8

Definition 2.2. Let M be a Riemannian manifold, then the operator

D 0 = d + δ : Λ even ( M ) Λ odd ( M ) E9

is called the Hodge‐de Rham operator. It has the property that it is a self‐conjugate operator, D 0 * = D 1 and D 1 * = D 0 . It is useful in studying the Laplacian to have a formula for the operator Δ = ( d + δ ) 2 and hence for D 0 * D 0 and D 1 * D 1 as well.

Let { e 1 , , e n } be an orthonormal moving frame defined on an open set U . Define as well the pair of operators

E j + = ω j + i ( e j ) : Λ * ( U ) Λ * ( U ) , E j = ω j i ( e j ) : Λ * ( U ) Λ * ( U ) E10

Lemma 2.1. The operators E j ± satisfy the following relations

E i + E j + + E j + E i + = 2 δ i j , E i + E j + E j E i + = 0, E i E j + E j E i = 2 δ i j E11

If M is a Riemannian manifold and : Γ ( T M ) × Γ ( T M ) Γ ( T M ) is a Levi‐Civita connection, then a connection on the space Λ * ( M ) , namely ( X , ω ) X ω , can also be defined such that

( X ω ) ( Y ) = X ( ω ( Y ) ) ω ( X Y ) , Y Γ ( T M )

The connection may be regarded as a first‐order derivative operator ( X , Y , ω ) D ( X , Y ) ω .

Definition 2.3. The second‐order derivative operator ( X , Y , ω ) D ( X , Y ) ω is defined to be

D ( X , Y ) ω = X Y ω X Y ω E12

In terms of the operator (Eq. (12)), define a second‐order differential operator Δ 0 : Λ * ( M ) Λ * ( M ) by

Δ 0 = i D ( e i , e i ) , E13

where { e i } 1 n is an orthonormal moving frame. The operator Δ 0 in Eq. (13) is referred to as the Laplace‐Beltrami operator.

Theorem 2.1. (Weitzenböck) Let M be a Riemannian manifold M with an associated orthonormal moving frame { e i } 1 n . The Laplace operator can be expressed as

Δ = ( d + δ ) 2 = Δ 0 1 8 i , j , k , s R i j k s E i + E j + E k E s + 1 4 R E14

In Eq. (14), R is the scalar curvature, R = i , j R i j i j and Δ 0 is the Laplace‐Beltrami operator (13).

The operator defined by Eq. (14) does not contain first‐order covariant derivatives and is of a type called a Schrödinger operator. Thus, Weitzenböck formula (14) implies the that Laplacian can be expressed in the form Δ = Δ 0 F and is an elliptic operator. The Schrödinger operator (14) can be used to define an operator that plays an important role in mathematical physics. The heat operator is defined to be

H = t + Δ E15

The crucial point for the theory of the heat operator is the existence of a fundamental solution. In fact, the Hodge theorem can be proved by making use of the fundamental solution.

Definition 2.4. Let M be a Riemannian manifold, π : E M is a vector bundle with connection. Let Δ 0 : Γ ( E ) Γ ( E ) be the Laplace‐Beltrami operator, which is defined by means of the Levi‐Civita connection on M and the connection on the vector bundle E . Let F : Γ ( E ) Γ ( E ) be a F ( M ) ‐linear map. Then, Δ = Δ 0 F is a Schrödinger operator. If a family of R ‐linear maps

G ( t , q , p ) : E p E q

with parameter t > 0 and q , p M satisfies the following three conditions, the family is called a fundamental solution of the heat operator (15) where E p = π 1 ( p ) . First, G ( t , q , p ) : E p E q is an R‐linear map of vector spaces and continuous in all variables t , q , p . Second, for a fixed w E p , let θ ( t , q ) = G ( t , q , p ) w , for all t > 0 , then θ has first and second continuous derivatives in t and q , respectively andsatisfies the heat equation, which for t > 0 is given by H θ ( t , q ) = 0 , which can be written as

( t + Δ q ) G ( t , q , p ) = 0 E16

where Δ q acts on the variable q . Finally, if φ is a continuous section of the vector bundle E , then

lim t 0 + M G ( t , q , p ) φ ( p ) d v p = φ ( q )

for all φ , where d v p is the volume measure with respect to the coordinates of p given in terms of the Riemannian metric.

Definition 2.5. Suppose a G 0 ( t , q , p ) is given. The following procedure taking G 0 ( t , q , p ) to G ( t , q , p ) is called the Levi algorithm:

K 0 ( t , q , p ) = ( t + Δ q ) G ( t , q , p ) , K m + 1 ( t , q , p ) = 0 t d τ M K 0 ( t τ , q , z ) K m ( τ , z , p ) d v z K ¯ ( t , q , p ) = m = 0 ( 1 ) m + 1 K m ( t , q , p ) , G ( t , q , p ) = G 0 ( t , q , p ) + 0 t d τ M G 0 ( t τ , q , z ) K ¯ ( τ , z , p ) d v z E17

The Cauchy problem can be formulated for the heat equation such that existence, regularity and uniqueness of solution can be established. The Hilbert‐Schmidt theorem can be invoked to develop a Fourier expansion theorem applicable to this Schrödinger operator.

Suppose Δ : Γ ( E ) Γ ( E ) is a self‐adjoint nonnegative Schrödinger operator, then there exists a set of C sections { ψ i } Γ ( E ) such that

ψ i , ψ j = M ψ i ( x ) , ψ j ( x ) d v x = δ i j

Moreover, denoting the completion of the inner product space Γ ( E ) by Γ ( E ) ¯ , the set { ψ i } is a complete set in Γ ( E ) ¯ , so for any ψ Γ ( E ) ¯ ,

ψ = i = 1 ψ , ψ i ψ i

Finally, the set { ψ i } satisfies the equation

Δ ψ i = λ i ψ i , T t ψ i = e t λ i ψ i

where λ i are the eigenvalues of Δ andform an increasing sequence: 0 λ 1 λ 2 where lim k λ k = .

Denote U ( t , q ) by ( T t ψ ) ( q ) when U ( 0, q ) = ψ ( q ) and T t satisfies the semigroup property and T t is a self‐adjoint, compact operator.

Theorem 2.2. Let G ( t , q , p ) be the fundamental solution of the heat operator (15), then

G ( t , q , p ) w = i = 1 e λ i t ψ i ( p ) , w ψ i ( q ) E18

with w E p holds in Γ ( E ) ¯ .

Proof: For fixed t > 0 and w E p , expand G ( t , q , p ) w in terms of eigenfunctions ψ i ( q ) ,

G ( t , q , p ) w = i = 1 σ i ( t , p , w ) ψ i ( q ) , σ i ( t , p , w ) = M ψ i ( q ) , G ( t , q , p ) w d v q

Differentiating with respect to t and using Δ ψ i = λ i ψ i , we get

t σ i ( t , p , w ) = M ψ i ( q ) , t G ( t , q , p ) w d v q = M ψ i ( q ) , Δ q G ( t , q , p ) w d v q = M Δ q ψ i ( q ) , G ( t , q , p ) w d v q = λ i M ψ i ( q ) , G ( t , q , p ) w d v q = λ i σ i ( t , p , w )

It follows from this that

σ i ( t , p , w ) = c i ( p , w ) e λ i t

and since σ i depend linearly on w , so c i ( p , w ) = c i ( p ) w , where c i ( p ) : E p R is a linear function. There exists c ˜ i ( p ) independent of w such that c i ( p ) w = c ˜ i ( p ) , w so that

G ( t , q , p ) w = i = 1 e λ i t ψ i ( q ) c ˜ i ( p ) , w

Consequently, for any β Γ ( E ) , we have

β ( q ) = lim t 0 M G ( t , q , p ) β ( p ) d v p = k = 1 ψ k ( q ) M c ˜ k ( p ) , β ( p ) d v p

Moreover, β ( q ) can also be expanded in terms of the ψ k basis set,

β ( q ) = k = 1 ψ k ( q ) M ψ k ( p ) , β ( p ) d v p

Upon comparing these last two expressions, it is clear that c ˜ k ( p ) = ψ k ( p ) for all k andwe are done.

One application of the heat equation method developed so far is to develop and give a proof of the Hodge theorem.

Theorem 2.3. Let M , E , Δ be defined as done already, then

1. H = { φ Γ ( E ) | Δ φ = 0 } is a finite‐dimensional vector space.

2. For any ψ Γ ( E ) , there is a unique decomposition of ψ as ψ = ψ 1 ψ 2 , where ψ 1 H and ψ 2 Δ ( Γ ( E ) ) .

The first part is a direct consequence of the expansion theorem and due to the fact H Δ ( Γ ( E ) ) , the decomposition is unique.

The Hodge theorem has many applications, but one in particular fits here. It is used in conjunction with the de Rham cohomology group H d R * ( M ) . Define

Z k ( M ) = ker { d : Λ k ( M ) Λ k + 1 ( M ) } { α Λ k ( M ) | d α = 0 } E19
B k ( M ) = Im { d : Λ k 1 ( M ) Λ k ( M ) } d ( Λ k 1 ( M ) ) E20

Since d 2 = 0 , it follows that B k ( M ) Z k ( M ) andthe k ‐th de Rham cohomology group of M is defined to be

H d R k ( M ) = Z k ( M ) / B k ( M ) E21

From Eq. (21), construct

H d R * ( M ) = k H d R k ( M ) E22

In 1935, Hodge claimed a theorem, which stated every element in H d R k ( M ) can be represented by a unique harmonic form α , one which satisfies both d α = 0 and δ α = 0 . Denote the set of harmonic forms as H k ( M ) .

Theorem 2.4. Let M be a Riemannian manifold of dimension n , then

H k ( M ) = ker { d + δ : Λ k ( M ) Λ * ( M ) } = ker { Δ : Λ k ( M ) Λ k ( M ) } E23

where Δ = ( d + δ ) 2 .

Proof: Since Δ = d δ + δ d , this implies that Δ ( Λ k ( M ) ) Λ k ( M ) andit is clear that

H k ( M ) ker { d + δ : Λ k ( M ) Λ * ( M ) } ker { Δ : Λ k ( M ) Λ * ( M ) } = ker { Δ : Λ k ( M ) Λ k ( M ) } . To finish the proof, it suffices to show that ker { Δ : Λ k ( M ) Λ k ( M ) } H k ( M ) . If α ker { Δ : Λ k ( M ) Λ k ( M ) } , that is Δ α = 0 , then

Δ α , α , = ( d + δ ) 2 α , α = ( d + δ ) α , ( d + δ ) α = d α , d α + δ α , δ α + 2 d α , δ α = d α , d α + δ α , δ α = 0

This implies that d α = 0 and δ α = 0 , hence α H k ( M ) .

Theorem 2.5. Let M be a Riemannian manifold of dimension n , then

1. H k ( M ) is a finite dimensional vector space for k = 0,1,2, , n .

2. There is an orthogonal decomposition of Λ k ( M ) as

Λ k ( M ) = H k ( M ) + d ( Λ k 1 ( M ) ) + δ ( Λ k + 1 ( M ) ) E24

Proof: By Theorem 2.1, Δ : Λ k ( M ) Λ k ( M ) is a Schrödinger operator, so the Hodge theorem applies. Thus H k ( M ) is of finite dimension, so the first holds. The second part of the Hodge theorem is Λ k ( M ) = H k ( M ) + Δ ( Λ k ( M ) ) . Since Δ ( Λ k ( M ) ) d ( Λ k 1 ( M ) ) + δ ( Λ k + 1 ( M ) ) , we have Λ k ( M ) = H k ( M ) + d ( Λ k 1 ( M ) ) + δ ( Λ k + 1 ( M ) ) . The three spaces in this decomposition are orthogonal to each other, so (ii) holds as well.

Theorem 2.6. (Duality theorem) For an oriented Riemannian manifold M of dimension n , the star isomorphism * : H k ( M ) H n k ( M ) induces an isomorphism

H d R k ( M ) H d R n k ( M ) E25

The k ‐th Betti number defined as b k ( M ) = dim H k ( M , R ) also satisfies b k ( M ) = b n k ( M ) for 0 k n .

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## 3. The Minakshisundaran‐Pleijel paramatrix

Let M be a Riemannian manifold with dimension n and E a vector bundle over M with an inner product and a metric connection. Here, the following formal power series is considered with a special transcendental multiplier e ρ 2 / 4 t and parameters ( t , p , q ) ( 0, ) × M × M , defined by

H ( t , q , p ) = 1 ( 4 π t ) n / 2 e ρ 2 / 4 t k = 0 t k u k ( p , q ) : E p E q E26

In Eq. (26), the function ρ = ρ ( p , q ) is the metric distance between p and q in M , E p = π 1 ( p ) is the fiber of E over p and u k ( p , q ) : E p E q are R ‐linear map.

It is the objective to find conditions for which Eq. (26) satisfies the heat equation or the following equality:

( t + Δ q ) H ( t , q , p ) w = 0 E27

To carry out this, a normal coordinate system denoted by { x 1 , , x n } is chosen in a neighborhood of point p and is centered at p . This means that if q is in this neighborhood about p , which has coordinates ( x 1 , , x n ) , then the function ρ ( p , q ) is

ρ ( p , q ) = x 1 2 + + x n 2 E28

In terms of these coordinates, we calculate the components of g ,

g i j = x i , x j , G = det ( g i j ) E29

and define the differential operator

^ = k = 1 n x k x k

The notion of the heat operator (15) on Eq. (26) is worked out one term at a time. First, the derivative with respect to t is calculated

t H ( t , p , q ) w = 1 ( 4 π t ) n / 2 e ρ 2 / 4 t { ( ρ 2 4 t 2 n 2 t ) k = 0 t k u k ( p , q ) w + k = 0 k t k 1 u k ( p , q ) w } = 1 ( 4 π t ) n / 2 e ρ 2 / 4 t k = 0 { ρ 2 4 t 2 n 2 t + k t } t k u k ( p , q ) w E30

It is very convenient to abbreviate the function appearing in front of the sum in Eq. (30) as follows:

Φ ( ρ ) = e ρ 2 / 4 t ( 4 π t ) n / 2 E31

Let { e 1 , , e n } be a frame that is parallel along geodesics passing through p and satisfies

e i ( p ) = x i | p E116

In terms of the function in Eq. (31), the operator Δ 0 acting on Eq. (26) is given as

Δ 0 H ( t , p , q ) w = ( Δ 0 Φ ) ( k = 0 t k u k ( p , q ) w ) + 2 a = 1 n ( e a Φ ) e a ( k = 0 t k u k ( p , q ) w ) + Φ Δ 0 ( k = 0 t k u k ( p , q ) w ) E32

The individual components of (32) can be calculated as follows; since Φ is a function e a Φ = e a Φ and so

e a Φ ( ρ ) = Φ ( ρ ) e a ( ρ ) , Δ 0 Φ = a { e a e a Φ ( ρ ) ( e a e a ) Φ ( ρ ) } = Φ ( ρ ) a ( e a ρ ) 2 + Φ ( ρ ) Δ 0 ρ , Φ ( ρ ) = ρ 2 t Φ ( ρ ) , Φ ( ρ ) = ( ρ 2 4 t 2 1 2 t ) Φ ( ρ ) E33

Consequently,

e a ρ = x a ρ , a ( e a ρ ) 2 = 1, Δ 0 ρ = n 1 ρ + 1 ρ ^ log G E117

and the Laplace‐Beltrami operator on the function Φ is given by

Δ 0 Φ = Φ ( ρ ) ( ( ρ 2 4 t 2 1 2 t ) 1 2 t ( n 1 ^ log G ) ) E34

Expression (34) goes into the first term on the right side of Eq. (32). The second term on the right‐hand side of (32) takes the form,

2 a = 1 n ( e a Φ ) e a ( k = 0 t k u k ( p , q ) w ) = 2 Φ ( ρ ) a = 1 n x a ρ e a ( k = 0 t k u k ( p , q ) w ) = ρ t Φ ( ρ ) ^ / ρ ( k = 0 t k u k ( p , q ) w ) E35

Substituting these results into (32), it follows that

Δ 0 H ( t , q , p ) = Φ ( ρ ) [ ρ 2 4 t 2 1 2 t 1 2 t ( n 1 ^ log G ) ρ t ^ / ρ + Δ 0 ] m = 0 t m u m ( p , q ) w E36

Combining Eq. (36) with the derivative of H with respect to t in Eq. (35), the following version of the heat equation results:

( t Δ 0 F ) H ( t , q , p ) w = Φ [ ( ^ + 1 4 G ^ G ) 1 t u 0 ( p , q ) w + k = 1 [ ( ^ + k + 1 4 G ^ G ) u k ( p , q ) w ( Δ 0 + F ) u k 1 ( p , q ) w t k 1 E37

This is summarized in the following Lemma.

Lemma 3.1. Heat equation (27) for H ( t , p , q ) is equivalent to

( ^ + k + 1 4 G ^ G ) u k ( p , q ) w = ( Δ 0 + F ) u k 1 ( p , q ) w E38

for all k = 0,1,2, and Eq. (38) is initialized with u 1 ( p , q ) = 0 .

In fact, for fixed p M and w E p , there always exists a unique solution to problem (Eq. (38)) over a small coordinate neighborhood about p .

Definition 3.1. Denote the solution of Eq. (38) by u ( p , q ) w , which depends linearly on w . Then, u m ( p , q ) : E p E q and the Minakshisundaram‐Pleijel parametrix for heat operator (Eq. 15) is defined by

H ( t , p , q ) = 1 ( 4 π t ) n / 2 e ρ 2 / 4 t m = 0 t m u m ( p , q ) : E p E q E39

Based on Eq. (39), the N ‐truncated parametrix is defined based on Eq. (39) to be

H N ( t , q , p ) = 1 ( 4 π t ) n / 2 e ρ 2 / 4 t m = 0 N t m u m ( p , q ) : E p E q E40

Theorem 3.1. Choose a smooth function ϕ : M × M M and let G 0 ( t , q , p ) = ϕ ( q , p ) H N ( t , q , p ) . Then G 0 ( t , q , p ) is a k ‐th initial solution of the heat operator (15), where k = N 2 n 4 and z is the greatest integer less than or equal to z .

Proof: Clearly, G 0 is a linear map of vector spaces andis continuous and C in all parameters. From the previous calculation, it holds that

( t Δ 0 F ) H N ( t , q , p ) w = 1 ( 4 π t ) n / 2 e ρ 2 / 4 t t N n 2 ( Δ 0 + F ) u N ( p , q ) w E41

and u N ( p , q ) is C with respect to p and q . Since t N n 2 e ρ 2 / 4 t is C k ( [ 0, ) × M × M ) , hence H ( φ ( p , q ) H N ( t , q , p ) ) C k ( [ 0, ) × M × M ) . Consider integrating G 0 against ψ ( s , β ) ,

M G 0 ( t , q , s ) ψ ( s , β ) d v s = m = 0 N t m M 1 ( 4 π t ) n / 2 e ρ 2 / 4 t ψ ( q , s ) u m ( s , q ) ψ ( s , β ) d v s E42

The integral of Eq. (42) over M can be broken up into an integral over Q q ( ò 2 ) = { s M | ρ ( q , s ) < ò / 2 } anda second integral over the set M M q ( ò 2 ) . On the latter set, the limit converges uniformly hence

lim t e ρ 2 / 4 t ( 4 π t ) n / 2 = 0

To estimate the remaining integral, choose a normal coordinate system at q and denote the integration coordinates as ( s 1 , , s n ) , then the integrand of Eq. (42) is given as

1 ( 4 π t ) n / 2 e | s | 2 / 4 t φ ( q , s ) u m ( s , q ) ψ ( s , β ) det s i , s j d s 1 d s n

Therefore, in the limit using Definition 2.4,

lim t 0 M ( ò / 2 ) 1 ( 4 π t ) n / 2 e ρ 2 / 4 t φ ( q , s ) u m ( s , q ) ψ ( s , β ) d v s = u m ( q , q ) ψ ( q , β )

This result implies that

lim t 0 M G 0 ( t , q , s ) ψ ( s , β ) d v s = m = 0 N lim t 0 t m u m ( q , q ) ψ ( q , β ) = ψ ( q , β ) u 0 ( q , q ) = ψ ( q , β ) E43

The convergence here is uniform.

There exists an asymptotic expansion for the heat kernel which is extremely useful and has several applications. It is one of the main intentions here to present this. An application of its use appears later.

Theorem 3.2. (Asymptotic expansion) Let M be a Riemannian manifold with dimension n and E a vector bundle over M with inner product and metric Riemannian connection. Let G ( t , q , p ) be the heat kernel or fundamental solution for heat operator (Eq. (15)) and (Eq. (39)) the MP parametrix. Then as t 0 , G ( t , p , p ) has the asymptotic expansion G ( t , p , p ) H ( t , p , p ) , that is, for any N > 0 , it is the case that

G ( t , p , p ) 1 ( 4 π t ) n / 2 m = 0 N t m u m ( p , p ) = O ( t N n 2 ) E44

and the symbol on the right‐hand side of Eq. (44) signifies a quantity ξ with the property that

lim t 0 ξ t N n 2 = 0

Proof: It suffices to prove the theorem for any large N . Let G 0 ( t , q , p ) = φ ( q , p ) H N ( t , q , p ) as in Theorem 3.2. The conclusion of the theorem is equivalent to the statement

G ( t , p , p ) G 0 ( t , p , p ) = O ( t N n 2 )

From the previous theorem and existence and regularity of the fundamental solution, the result G of Levi iteration initialized by G 0 is exactly the fundamental solution. Equality (Eq. (41)) means that there exists a constant A such that for any t ( 0, T ) ,

| K 0 ( t , q , p ) | = | ( t + Δ ) G 0 ( t , q , p ) | A t N n 2

Let v ( M ) be the volume of the manifold M . Using this result, the following upper bound is obtained

| K 1 ( t , q , p ) | 0 t d τ M | K 0 ( t τ , q , s ) K 0 ( τ , s , p ) | d v s 0 t [ A 2 ( t τ ) N n 2 τ N n 2 v ( M ) ] d τ 0 t A 2 T N n 2 τ N n 2 v ( M ) d τ A B t N n 2 + 1 N n 2 + 1

We have set B = A T N n 2 v ( M ) . Exactly the same procedure applies to | K 2 ( t , q , p ) | . Based on the pattern established this way, induction implies that the following bound results

| K m ( t , q , p ) | A B m t N n 2 + m ( N n 2 + 1 ) ( N n 2 + 2 ) ( N n 2 + m ) A B m t m m ! t N n 2

The formula for Levi iteration yields upon summing this over m the following upper bound

| K ˜ ( t , q , p ) | m = 0 | K m ( t , q , p ) | A e B t t N n 2

Using this bound, the required estimate is obtained,

| G ( t , q , p ) G 0 ( t , q , p ) | | 0 t d τ M d v z G 0 ( t τ , q , z ) K ˜ ( τ , z , p ) | 0 t d τ M e ρ 2 / 4 ( t τ ) ( 4 π ( t τ ) ) n / 2 A e B τ τ N n 2 d v s M n A e B t 0 t τ N n 2 d τ v ( M ) = 1 N n 2 + 1 M n A e B t v ( M ) t N n 2 + 1

This finishes the proof.

Now if all the Hodge theorem is used, formal expressions for the index can be obtained. Suppose D : Γ ( E ) Γ ( F ) is an operator such that D * D and D D * are Schrödinger operators and D * is the adjoint of D . Suppose the operators D * D : Γ ( E ) Γ ( E ) and D D * : Γ ( F ) Γ ( E ) are defined, so they are self‐adjoint and have nonnegative real eigenvalues. Then the spaces Γ μ ( E ) and Γ μ ( F ) can be defined this way

Γ μ ( E ) = { φ Γ ( E ) | D * D φ = μ φ } , Γ μ ( F ) = { φ Γ ( F ) | D D * φ = μ φ } E45

For any m > 0 , the dimensions of the spaces in (44) are finite and moreover,

Γ 0 ( E ) = ker { D : Γ ( E ) Γ ( F ) } , Γ 0 ( F ) = ker { D * : Γ ( F ) Γ ( E ) }

Consequently, an expression for the index Ind ( D ) can be obtained from Eq. (45) as follows

Ind D = dim ker D dim ker D * = dim Γ 0 ( E ) dim Γ 0 ( F )

Definition 3.2. For the Schrödinger operator Δ , let e t Δ : Γ ( E ) Γ ( E ) , for t > 0 be defined as

( e t Δ φ ) ( q ) = M G ( t , q , p ) φ ( p ) d v p E46

where G ( t , q , p ) is the fundamental solution of heat operator (Eq. (15)).

Let 0 λ 1 λ 2 be the eigenvalues of the operator Δ and { ψ 1 , ψ 2 , } the corresponding eigenfunctions. Intuitively, the trace of e t Δ is defined as

t r e t Δ = k = 1 e t Δ ψ k , ψ k E47

This is clearly k e λ k t or μ e t μ dim Γ μ ( E ) , so the definition of t r is well‐defined if and only if

k e λ k t < E48

Theorem 3.3. For any p , q M , let { e 1 ( p ) , , e N ( p ) } and { f 1 ( q ) , , f N ( q ) } be orthonormal bases on E p and E q , respectively, then the following two results hold for t > 0 ,

( a ) M a , b = 1 N G ( t , q , p ) e a ( p ) , f b ( q ) 2 d v q d v p < , ( b ) k = 1 e 2 λ k t < M a , b = 1 N G ( t , q , p ) e a ( p ) , f b ( q ) 2 d v q d v p < E49

Proof: When t > 0 , G ( t , q , p ) is continuous and hence satisfies (a). For and w Γ ( E ) , Theorem 2.5 yields the following expansion for G ( t , q , p ) Γ ( E ) ¯ , hence the Parseval equality yields

M | G ( t , q , p ) w | 2 d v q = k = 1 e 2 λ k t ψ k ( p ) , w 2

Replacing w by the basis element e a ( p ) , this implies that

a = 1 N M | G ( t , q , p ) e a ( p ) | 2 d v q = a = 1 N k = 1 e 2 λ k t ψ k ( p ) , e a ( p ) 2 = k = 1 a = 1 N e 2 λ k t ψ k ( p ) , e a ( p ) 2 = k = 1 e 2 λ k t ψ k ( p ) , ψ k ( p )

Then for any m , it follows that

k = 1 m e 2 λ k t = k = 1 m M e 2 λ k t ψ k ( p ) , ψ k ( p ) d v p M k = 1 e 2 λ k t ψ k ( p ) , ψ k ( p ) d v p = M d v p M a = 1 N | G ( t , q , p ) e a ( p ) | 2 d v q = M M a , b = 1 N G ( t , q , p ) e a ( p ) , f b ( q ) 2 d v q d v p <

Theorem 3.4. For any t > 0 ,

tr ( e t Δ ) = M t r G ( t , p , p ) d v p E50

Proof: From Theorem 2.2, it follows that

t r G ( t , p , p ) = a = 1 N G ( t , p , p ) e a ( p ) , e a ( p ) = a = 1 N k = 1 e t λ k ψ k ( p ) e a ( p ) ψ k ( p ) , e a ( p ) = a = 1 N k = 1 e t λ k ψ k ( p ) , e a ( p ) 2 = k = 1 e t λ k ψ k ( p ) , ψ k ( p ) 2

Integrating this on both sides, it is found that

M tr G ( t , p , p ) d v p = M k = 1 e t λ k ψ k ( p ) , ψ k ( p ) 2 d v p = k = 1 e t λ k = tr ( e t Δ )

Note that Eq. (48) is a series with positive terms which converges uniformly as t . Therefore,

lim t tr e t Δ = k = 1 lim t e t λ k = dim Γ 0 ( E ) E51

In fact, as t 0 , the equality

G ( t , p , p ) = 1 ( 4 π t ) n / 2 + O ( 1 t n / 2 )

and the previous theorem imply that lim t 0 tr e t Δ = .

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## 4. An application of the expansions: the Gauss Bonnet theorem

As far as Ind ( D ) is concerned, it is the case for all t > 0 that,

Ind ( D ) = tr e t D * D tr e t D D * = M tr G + ( t , p , p ) d v p M tr G ( t , p , p ) d v p

by Theorem 3.5, where G ± ( t , p , p ) are the fundamental solutions of t + D * D and t + D D * . As t 0 , Theorem 3.2 assumes the form

G ± ( t , p , p ) H ± ( t , p , p ) = 1 ( 4 π t ) n / 2 m = 0 t m u ± m ( p , p )

Lemma 4.1. Let { λ i } be the spectrum of the Laplacian on zero‐forms, or functions, on M . Then,

k e λ k t = 1 ( 4 π t ) n / 2 k = 0 M u k ( x , x ) d v x E52

Proof:

k e λ k t = M tr G ( t , x , x ) d v x = 1 ( 4 π t ) n / 2 k ( M u k ( x , x ) d v x ) t k

The spectrum of the Laplacian on functions characterizes a lot of interesting geometric information. Note that Eq. (52) can be written as

i e λ i t 1 ( 4 π t ) n / 2 k = 0 a k t k , a k = M u k ( x , x ) d v x

and the trace does not appear in the case of functions. The superscript on the Laplacian Δ p denotes the form degree acted upon andsimilarly on other objects throughout this section.

Two Riemannian manifolds are said to be isospectral if the eigenvalues of their Laplacians on functions counted with multiplicities coincide.

Corollary 4.1. Let M and N be compact isospectral Riemannian manifolds. Then M and N have the same dimension and the same volume.

Proof: Let { λ i } denote the spectrum of both M and N with dim M = m and dim N = n . Then it follows that

1 ( 4 π t ) m / 2 k = 0 ( M u k M ( p , p ) d v p ) t k = i = 0 e λ i t = 1 ( 4 π t ) n / 2 k = 0 ( N u k N ( q , q ) d v q ) t k

This implies that m = n , which in turn implies that

1 ( 4 π t ) m / 2 [ M u 0 M ( p , p ) d v p N u N ( q , q ) d v q ] = 1 ( 4 π t ) m / 2 k = 1 ( M u k M ( p , p ) d v p N u N ( q , q ) d v q ) t k

Since the right‐hand side of the equation depends on t , but the left‐hand side does not, this result implies that

M u 0 M ( p , p ) d v p = N u 0 N ( q , q ) d v q E53

Iterating this argument leads to the set of equations

M u k M ( p , p ) d v p = N u k N ( q , q ) d v q E54

for all k > 0 . In particular, since u 0 = 1 , Eq. (53) leads to the conclusion vol ( M ) = vol ( N ) .

The proof illustrates that in fact there exist an infinite sequence of obstructions to claiming that two manifolds are isospectral, namely the set of integrals M u k d v p . The first integral contains basic geometric information. It is then natural to investigate the other integrals in sequence as well. Recall that R p , R p , denote the covariant derivatives of the curvature tensor at p . A polynomial P in the curvature and its covariant derivatives is called universal if its coefficients depend only on the dimension of M . The notation P ( R p , R p , , k R p ) is used to denote a polynomial in the components of the curvature tensor and its covariant derivatives calculated in a normal Riemannian coordinate chart at p . The following theorem will not be proved, but it will be used shortly.

Theorem 4.2. On a manifold of dimension n ,

u 1 ( p , p ) = P 1 n ( R p ) , u k ( p , p ) = P k n ( R p , R p , , 2 k 2 R p ) , k 2 E55

for some universal polynomials P k n .

Thus, P 1 n is a linear function with no constant term and u 1 ( p , p ) is a linear function of the components of the curvature tensor at p , with no covariant derivative terms. The only linear combination of curvature components that produces a well‐defined function u 1 ( p , p ) on a manifold is the scalar curvature R ( p ) = R i j i j andso there exists a constant C such that u 1 ( p , p ) = C R ( p ) .

Theorem 4.3.

u 1 ( p , p ) = 1 6 R ( p ) E56

Proof: The proof amounts to noticing that P 1 n is a universal polynomial, so it suffices to compute C over one kind of manifold. A good choice is to integrate over S n with the standard metric and work it out explicitly in normal coordinates. It is found that u 1 ( p , p ) = n ( n 1 ) / 6 andit is known that R ( p ) = n ( n 1 ) for all p S n andthis implies Eq. (56).

The large t or long‐time behavior of the heat operator for the Laplacian on differential forms is then controlled by the topology of the manifold through the means of the de Rham cohomology. The small t or short‐time behavior is controlled by the geometry of the asymptotic expansion. The combination of topological information has a geometric interpretation. This is made explicit by means of the Chern‐Gauss‐Bonnet theorem. The two‐dimensional version of this theorem will be developed here.

These results can be summarized by the elegant formula

k = 0 e λ k t = 1 ( 4 π t ) n / 2 { v ( M ) + 1 6 M R ( x ) d v x t + O ( t 2 ) }

where v ( M ) is the volume of M .

Suppose that λ is positive and here we let E λ p denote the possibly trivial eigenspace of Δ on p ‐forms. If ω E λ p then it follows that Δ p + 1 d ω = d Δ p ω = λ d ω , hence d ω E λ p + 1 . Thus, a well‐defined sequential ordering of the spaces can be established. If ω E λ p has the property that d ω = 0 , then λ ω = Δ p ω = ( δ d + d δ ) ω = d δ ω . Therefore, since λ 0 , it is found that ω = d ( 1 λ δ ω ) . Thus, the sequence 0 E λ 0 d d E λ n 0 is exact. Since the operator d + δ is an isomorphism on k E λ 2 k , it follows that

s ( 1 ) s dim E λ s = 0 E57

Theorem 4.4. Let { λ i s } be the spectrum of the operator Δ , then

s ( 1 ) s i e λ i s t = s ( 1 ) s dim ker Δ s . E58

Proof: By (57),

s ( 1 ) s k e λ k s t = s ( 1 ) s e λ i t

The sum on the right is only over eigenvalues such that λ i p = 0 and so

e λ i p t = dim ker Δ p .

This has the consequence that

p ( 1 ) p tr e t Δ = p ( 1 ) p k e λ k p t E59

is independent of the parameter t . This means that its large or long t behavior is the same as its short or small t behavior. To put it another way, the long‐time behavior of tr e t Δ is given by the de Rham cohomology, while the short‐time behavior is dictated by the geometry of the manifold. Using the definition of the Euler characteristic, it follows that

χ ( M ) = p ( 1 ) p dim H d H p ( M ) = p ( 1 ) p dim ker Δ p = p ( 1 ) p tr e t Δ p = p ( 1 ) p M tr G ( t , x , x ) d v x E60

From the asymptotic expansion theorem, the following expression for χ ( M ) results

χ ( M ) = 1 ( 4 π t ) n / 2 k = 0 ( M s = 0 n tr u k s ( x , x ) d v x ) t k E61

The u k s in Eq. (61) are the coefficients in the asymptotic expansion for tr ( e t Δ s ) . Since χ ( M ) is independent of t , only the constant or t ‐independent term on the right‐hand side of Eq. (61) can be nonzero. This implies the following important theorem.

Theorem 4.5. If the dimension of M is even, then

1 ( 4 π ) n / 2 M s = 0 n ( 1 ) s tr u k s ( x , x ) d v x = { 0, k n 2 ; χ ( M ) , k = n 2 . E62

Theorem 4.6. (Gauss‐Bonnet) Let M be a closed oriented manifold with Gaussian curvature K and area measure d a M , then

χ ( M ) = 1 2 π M K d a M E63

Proof: By the last theorem and the fact that tr u k p ( x , x ) = tr u k p 1 ( x , x ) , it follows that

χ ( M ) = 1 4 π M p = 0 2 ( 1 ) p tr u 1 p d a M = 1 4 π M ( tr u 1 0 tr u 1 1 + tr u 1 2 ) d a M = 1 4 π M ( 2 tr u 1 0 tr u 1 1 ) d a M = 1 4 π M ( 2 3 K tr u 1 1 ) d a M E64

since the scalar curvature is two times the Gaussian. Now it must be that tr u 1 1 ( x , x ) = C R ( x ) = 2 C K ( x ) , for some constant C . The standard sphere S 2 has Gaussian curvature one andso C can be calculated from Eq. (64),

2 = 1 2 π S 2 ( 1 3 C ) d a M = 1 2 π ( 1 3 C ) ( 4 π )

Therefore, C = 2 / 3 and putting all of these results into Eq. (64), Eq. (62) results.

As an application of this theorem, note that the calculation of u 1 gives another topological obstruction to manifolds having the same spectrum.

Theorem 4.7. Let ( M , g ) and ( N , h ) be compact isospectral surfaces, then M and N are diffeomorphic.

Proof: As noted in Corollary 4.1,

M u 1 M ( x , x ) d v x = N u 1 N ( y , y ) d v y

On a surface, the scalar curvature is twice the Gaussian curvature, so by the Gauss‐Bonnet theorem,

6 π χ ( M ) = M u 1 M ( x , x ) d v x = N u 1 N ( y , y ) d v y = 6 π χ ( N ) E65

However, oriented surfaces with the same Euler characteristic are diffeomorphic.

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## 5. Summary and outlook

The heat equation approach has been seen to be quite deep, leading both to the Hodge theorem and also to a proof of the Gauss‐Bonnet theorem. Moreover, it is clear from the asymptotic development that there is a generalization of this theorem to higher dimensions. The four‐dimensional Chern‐Gauss‐Bonnet integrand is given by the invariant 1 32 π 2 { K 2 4 | ρ r | 2 + | R | 2 } , where K is the scalar curvature, | ρ r | 2 is the norm of the Ricci tensor, | R | 2 is the norm of the total curvature tensor andthe signature is Riemannian. This comes up in physics especially in the study of Einstein‐Gauss‐Bonnet gravity where this invariant is used to get the associated Euler‐Lagrange equations.

Let R i j k l be the components of the Riemann curvature tensor relative to an arbitrary local frame field { e i } for the tangent bundle T M and adopt the Einstein summation convention. Let m = 2 s be even, then the Pfaffian E m ( g ) is defined to be

E m ( g ) = 1 ( 8 π ) s s ! R i 1 i 2 j 2 j 1 R i 2 s 1 i 2 s j 2 s j 2 s 1 g ( e i 1 e i 2 s , e j 1 e j 2 s ) E66

The Euler characteristic χ ( M ) of any compact manifold of odd dimension without boundary vanishes. Only the even dimensional case is of interest.

Theorem 5.1. Let ( M , g ) be a compact Riemannian manifold without boundary of even dimension m . then

χ ( M ) = M E m ( g ) d v M E67

This was proved first by Chern, but of greater significance here, this can be deduced from the heat equation approach that has been introduced here. There is a proof by Patodi [18], but there is no room for it now. It should be hoped that more interesting results will come out in this area as well in the future.

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Written By

Paul Bracken

Submitted: 28 April 2016 Reviewed: 29 November 2016 Published: 18 January 2017