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Mathematics » "Manifolds - Current Research Areas", book edited by Paul Bracken, ISBN 978-953-51-2872-4, Print ISBN 978-953-51-2871-7, Published: January 18, 2017 under CC BY 3.0 license. © The Author(s).

# Spectral Theory of Operators on Manifolds

By Paul Bracken
DOI: 10.5772/67095

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# Spectral Theory of Operators on Manifolds

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].

## 2. Geometrical preliminaries

For an n ‐dimensional Riemannian manifold M , an orthonormal moving frame {e1,,en} can be chosen with {ω1,,ωn} the accompanying dual coframe which satisfy

 ωi(ej)=δij,  i,j=1,…,n (1)

It is then possible to define a system of one‐forms ωij and two‐forms Ωij by solving the equations,

 ∇Xei=∑j ωji(X) ej,  R(X,Y)ei=∑j Ωji(X,Y)ej (2)

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

 ωji(ek)=∑a〈ωaj(ek)ea,ei〉g=〈∇ekej,ei〉g=Γkji (3)
 Ωij(ek,es)=∑a 〈Ωaj(ek,es)ea,ei〉g=〈 R(ek,es)ej,ei〉g=Rksji (4)

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

 〈〈α,β〉〉=∫M 〈α,β〉g dvM (5)

where if (x1,,xm) is a system of local coordinates,

dvM=det(gij)dx1dxm

is the Riemannian measure on M . Clearly, α,β is linear with respect to α , β and α,α0 with equality if and only if α=0 . Hodge introduced a star homomorphism * :Λk(M)Λnk(M) , which is defined next.

Definition 2.1. (i) For ω=i1<<ikfi1ikωi1ωik , define

*ω=i1<<ikj1<<jnkfi1ikò(i1,,ik,j1,,jnk)ωj1ωjnk,

where ò is 1 , 1 , or 0 depending on whether (i1,,ik,j1,,jnk) 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)nk+n+1*d*:Λk(M)→Λk−1(M) (6)

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

 Λeven(M)=⊕even Λk(M),  Λodd(M)=⊕odd Λk(M) (7)

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

 D0=d+δ:Λeven(M)→Λodd(M), D1=d+δ:Λodd(M)→Λeven(M) (8)

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

 D0=d+δ:Λeven(M)→Λodd(M) (9)

is called the Hodge‐de Rham operator. It has the property that it is a self‐conjugate operator, D0*=D1 and D1*=D0 . It is useful in studying the Laplacian to have a formula for the operator Δ=(d+δ)2 and hence for D0*D0 and D1*D1 as well.

Let {e1,,en} be an orthonormal moving frame defined on an open set U . Define as well the pair of operators

 Ej+=ωj∧⋅+i(ej):Λ*(U)→Λ*(U),  Ej−=ωj∧⋅−i(ej):Λ*(U)→Λ*(U) (10)

Lemma 2.1. The operators Ej± satisfy the following relations

 Ei+Ej++Ej+Ei+=2δij, Ei+Ej−+Ej−Ei+=0, Ei−Ej−+Ej−Ei−=−2δij (11)

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

(Xω)(Y)=X(ω(Y))ω(XY),YΓ(TM)

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∇Yω−∇∇XYω (12)

In terms of the operator (Eq. (12)), define a second‐order differential operator Δ0:Λ*(M)Λ*(M) by

 Δ0=∑i D(ei,ei), (13)

where {ei}1n is an orthonormal moving frame. The operator Δ0 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 {ei}1n . The Laplace operator can be expressed as

 Δ=(d+δ)2=−Δ0−18 ∑i,j,k,s Rijks Ei+Ej+Ek−Es−+14R (14)

In Eq. (14), R is the scalar curvature, R=i,jRijij and Δ0 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=Δ0F 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 (15)

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:EM is a vector bundle with connection. Let Δ0: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=Δ0F is a Schrödinger operator. If a family of R ‐linear maps

G(t,q,p):EpEq

with parameter t>0 and q,pM satisfies the following three conditions, the family is called a fundamental solution of the heat operator (15) where Ep=π1(p) . First, G(t,q,p):EpEq is an R‐linear map of vector spaces and continuous in all variables t,q,p . Second, for a fixed wEp , 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 Hθ(t,q)=0 , which can be written as

 (∂∂t+Δq)G(t,q,p)=0 (16)

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

limt0+MG(t,q,p)φ(p)dvp=φ(q)

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

Definition 2.5. Suppose a G0(t,q,p) is given. The following procedure taking G0(t,q,p) to G(t,q,p) is called the Levi algorithm:

 K0(t,q,p)=(∂∂t+Δq)G(t,q,p),Km+1(t,q,p)=∫0t dτ∫M K0(t−τ,q,z)Km(τ,z,p) dvzK¯(t,q,p)=∑m=0∞ (−1)m+1Km(t,q,p),G(t,q,p)=G0(t,q,p)+∫0t dτ∫M G0(t−τ,q,z)K¯(τ,z,p) dvz (17)

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 {ψi}Γ(E) such that

ψi,ψj=Mψi(x),ψj(x)dvx=δij

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

ψ=i=1ψ,ψiψi

Finally, the set {ψi} satisfies the equation

Δψi=λiψi,Ttψi=etλiψi

where λi are the eigenvalues of Δ andform an increasing sequence: 0λ1λ2 where limkλk= .

Denote U(t,q) by (Ttψ)(q) when U(0,q)=ψ(q) and Tt satisfies the semigroup property and Tt 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∞ eλit〈 ψi(p),w〉ψi(q) (18)

with wEp holds in Γ(E)¯ .

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

G(t,q,p)w=i=1σi(t,p,w)ψi(q),σi(t,p,w)=Mψi(q),G(t,q,p)wdvq

Differentiating with respect to t and using Δψi=λiψi , we get

tσi(t,p,w)=Mψi(q),tG(t,q,p)wdvq=Mψi(q),ΔqG(t,q,p)wdvq=MΔqψi(q),G(t,q,p)wdvq=λiMψi(q),G(t,q,p)wdvq=λiσi(t,p,w)

It follows from this that

σi(t,p,w)=ci(p,w)eλit

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

G(t,q,p)w=i=1eλitψi(q)c˜i(p),w

Consequently, for any βΓ(E) , we have

β(q)=limt0MG(t,q,p)β(p)dvp=k=1ψk(q)Mc˜k(p),β(p)dvp

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

β(q)=k=1ψk(q)Mψk(p),β(p)dvp

Upon comparing these last two expressions, it is clear that c˜k(p)=ψk(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={φΓ(E)|Δφ=0} is a finite‐dimensional vector space.

2. For any ψΓ(E) , there is a unique decomposition of ψ as ψ=ψ1ψ2 , where ψ1H and ψ2Δ(Γ(E)) .

The first part is a direct consequence of the expansion theorem and due to the fact HΔ(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 HdR*(M) . Define

 Zk(M)=ker{d:Λk(M)→Λk+1(M)}≡{α∈Λk(M)| dα=0} (19)
 Bk(M)= Im {d:Λk−1(M)→Λk(M)}≡d(Λk−1(M)) (20)

Since d2=0 , it follows that Bk(M)Zk(M) andthe k ‐th de Rham cohomology group of M is defined to be

 HdRk(M)=Zk(M)/Bk(M) (21)

From Eq. (21), construct

 HdR*(M)=⊕k HdRk(M) (22)

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

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

 Hk(M)=ker {d+δ:Λk(M)→Λ*(M)}=ker {Δ:Λk(M)→Λk(M)} (23)

where Δ=(d+δ)2 .

Proof: Since Δ=dδ+δd , this implies that Δ(Λk(M))Λk(M) andit is clear that

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

Δα,α,=(d+δ)2α,α=(d+δ)α,(d+δ)α=dα,dα+δα,δα+2dα,δα=dα,dα+δα,δα=0

This implies that dα=0 and δα=0 , hence αHk(M) .

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

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

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

 Λk(M)=Hk(M)+d(Λk−1(M))+δ(Λk+1(M)) (24)

Proof: By Theorem 2.1, Δ:Λk(M)Λk(M) is a Schrödinger operator, so the Hodge theorem applies. Thus Hk(M) is of finite dimension, so the first holds. The second part of the Hodge theorem is Λk(M)=Hk(M)+Δ(Λk(M)) . Since Δ(Λk(M))d(Λk1(M))+δ(Λk+1(M)) , we have Λk(M)=Hk(M)+d(Λk1(M))+δ(Λk+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 *:Hk(M)Hnk(M) induces an isomorphism

 HdRk(M)≃HdRn−k(M) (25)

The k ‐th Betti number defined as bk(M)=dimHk(M,R) also satisfies bk(M)=bnk(M) for 0kn .

## 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 eρ2/4t and parameters (t,p,q)(0,)×M×M , defined by

 H∞(t,q,p)=1(4πt)n/2e−ρ2/4t ∑k=0∞ tk uk(p,q):Ep→Eq (26)

In Eq. (26), the function ρ=ρ(p,q) is the metric distance between p and q in M , Ep=π1(p) is the fiber of E over p and uk(p,q):EpEq are R ‐linear map.

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

 (∂∂t+Δq)H∞(t,q,p)w=0 (27)

To carry out this, a normal coordinate system denoted by {x1,,xn} 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 (x1,,xn) , then the function ρ(p,q) is

 ρ(p,q)=x12+⋯+xn2 (28)

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

 gij=〈∂∂xi,∂∂xj〉,  G=det(gij) (29)

and define the differential operator

^=k=1nxkxk

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

 ∂∂tH∞(t,p,q)w=1(4πt)n/2e−ρ2/4t{(ρ24t2−n2t)∑k=0∞ tk uk(p,q)w+∑k=0∞ ktk−1uk(p,q)w}=1(4πt)n/2 e−ρ2/4t∑k=0∞ {ρ24t2−n2t+kt} tkuk(p,q)w (30)

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

 Φ(ρ)=e−ρ2/4t(4πt)n/2 (31)

Let {e1,,en} be a frame that is parallel along geodesics passing through p and satisfies

 ei(p)=∂∂xi|p (116)

In terms of the function in Eq. (31), the operator Δ0 acting on Eq. (26) is given as

 Δ0H∞(t,p,q)w=(Δ0 Φ)⋅(∑k=0∞ tkuk(p,q)w)+2∑a=1n (eaΦ)⋅∇ea(∑k=0∞ tkuk(p,q)w)+Φ⋅Δ0(∑k=0∞ tkuk(p,q)w) (32)

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

 eaΦ(ρ)=Φ′(ρ)ea(ρ),Δ0Φ=∑a{ea eaΦ(ρ)−(∇eaea)Φ(ρ)}=Φ″(ρ)⋅∑a(eaρ)2+Φ′(ρ)⋅Δ0ρ,Φ′(ρ)=−ρ2tΦ(ρ),Φ″(ρ)=(ρ24t2−12t)Φ(ρ) (33)

Consequently,

 eaρ=xaρ,  ∑a(eaρ)2=1,  Δ0ρ=n−1ρ+1ρ∂^ logG (117)

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

 Δ0 Φ=Φ(ρ)((ρ24t2−12t)−12t(n−1−∂^logG)) (34)

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=1n (eaΦ)⋅∇ea (∑k=0∞ tkuk(p,q)w)=2Φ′(ρ) ∑a=1n xaρ⋅∇ea(∑k=0∞ tkuk(p,q)w)=−ρtΦ(ρ)∇∂^/ρ(∑k=0∞ tkuk(p,q)w) (35)

Substituting these results into (32), it follows that

 Δ0 H∞(t,q,p)=Φ(ρ) [ρ24t2−12t−12t(n−1−∂^logG)−ρt∇∂^/ρ+Δ0]∑m=0∞ tmum(p,q)w (36)

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

 (∂∂t−Δ0−F)H∞(t,q,p)w=Φ[(∇∂^+14G∂^G)⋅1tu0(p,q)w+∑k=1∞ [(∇∂^+k+14G∂^G)uk(p,q)w−(Δ0+F)uk−1(p,q)wtk−1 (37)

This is summarized in the following Lemma.

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

 (∇∂^+k+14G∂^G) uk(p,q)w=(Δ0+F) uk−1(p,q)w (38)

for all k=0,1,2, and Eq. (38) is initialized with u1(p,q)=0 .

In fact, for fixed pM and wEp , 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, um(p,q):EpEq and the Minakshisundaram‐Pleijel parametrix for heat operator (Eq. 15) is defined by

 H∞(t,p,q)=1(4πt)n/2e−ρ2/4t ∑m=0∞ tmum(p,q):Ep→Eq (39)

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

 HN(t,q,p)=1(4πt)n/2e−ρ2/4t ∑m=0N tmum(p,q):Ep→Eq (40)

Theorem 3.1. Choose a smooth function ϕ:M×MM and let G0(t,q,p)=ϕ(q,p)HN(t,q,p) . Then G0(t,q,p) is a k ‐th initial solution of the heat operator (15), where k=N2n4 and z is the greatest integer less than or equal to z .

Proof: Clearly, G0 is a linear map of vector spaces andis continuous and C in all parameters. From the previous calculation, it holds that

 (∂∂t−Δ0−F)HN(t,q,p)w=−1(4πt)n/2e−ρ2/4ttN−n2(Δ0+F)uN(p,q)w (41)

and uN(p,q) is C with respect to p and q . Since tNn2eρ2/4t is Ck([0,)×M×M) , hence H(φ(p,q)HN(t,q,p))Ck([0,)×M×M) . Consider integrating G0 against ψ(s,β) ,

 ∫M G0(t,q,s)ψ(s,β) dvs=∑m=0N tm∫M 1(4πt)n/2e−ρ2/4tψ(q,s)um(s,q)ψ(s,β) dvs (42)

The integral of Eq. (42) over M can be broken up into an integral over Qq(ò2)={sM|ρ(q,s)<ò/2} anda second integral over the set MMq(ò2) . On the latter set, the limit converges uniformly hence

limteρ2/4t(4πt)n/2=0

To estimate the remaining integral, choose a normal coordinate system at q and denote the integration coordinates as (s1,,sn) , then the integrand of Eq. (42) is given as

1(4πt)n/2e|s|2/4tφ(q,s)um(s,q)ψ(s,β)detsi,sjds1dsn

Therefore, in the limit using Definition 2.4,

limt0M(ò/2)1(4πt)n/2eρ2/4tφ(q,s)um(s,q)ψ(s,β)dvs=um(q,q)ψ(q,β)

This result implies that

 limt→0 ∫M G0(t,q,s)ψ(s,β) dvs=∑m=0N limt→0 tmum(q,q)ψ(q,β)=ψ(q,β)u0(q,q)=ψ(q,β) (43)

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 t0 , 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(4πt)n/2 ∑m=0N tmum(p,p)=O(tN−n2) (44)

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

limt0ξtNn2=0

Proof: It suffices to prove the theorem for any large N . Let G0(t,q,p)=φ(q,p)HN(t,q,p) as in Theorem 3.2. The conclusion of the theorem is equivalent to the statement

G(t,p,p)G0(t,p,p)=O(tNn2)

From the previous theorem and existence and regularity of the fundamental solution, the result G of Levi iteration initialized by G0 is exactly the fundamental solution. Equality (Eq. (41)) means that there exists a constant A such that for any t(0,T) ,

|K0(t,q,p)|=|(t+Δ)G0(t,q,p)|AtNn2

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

|K1(t,q,p)|0tdτM|K0(tτ,q,s)K0(τ,s,p)|dvs0t[A2(tτ)Nn2τNn2v(M)]dτ0tA2TNn2τNn2v(M)dτABtNn2+1Nn2+1

We have set B=ATNn2v(M) . Exactly the same procedure applies to |K2(t,q,p)| . Based on the pattern established this way, induction implies that the following bound results

|Km(t,q,p)|ABmtNn2+m(Nn2+1)(Nn2+2)(Nn2+m)ABmtmm!tNn2

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

|K˜(t,q,p)|m=0|Km(t,q,p)|AeBttNn2

Using this bound, the required estimate is obtained,

|G(t,q,p)G0(t,q,p)||0tdτMdvzG0(tτ,q,z)K˜(τ,z,p)|0tdτMeρ2/4(tτ)(4π(tτ))n/2AeBττNn2dvsMnAeBt0tτNn2dτv(M)=1Nn2+1MnAeBtv(M)tNn2+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 DD* are Schrödinger operators and D* is the adjoint of D . Suppose the operators D*D:Γ(E)Γ(E) and DD*:Γ(F)Γ(E) are defined, so they are self‐adjoint and have nonnegative real eigenvalues. Then the spaces Γμ(E) and Γμ(F) can be defined this way

 Γμ(E)={φ∈Γ(E)|D*Dφ=μφ},  Γμ(F)={φ∈Γ(F)|DD*φ=μφ} (45)

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

Γ0(E)=ker{D:Γ(E)Γ(F)},Γ0(F)=ker{D*:Γ(F)Γ(E)}

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

IndD=dimkerDdimkerD*=dimΓ0(E)dimΓ0(F)

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

 (e−tΔφ)(q)=∫M G(t,q,p)φ(p)dvp (46)

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

Let 0λ1λ2 be the eigenvalues of the operator Δ and {ψ1,ψ2,} the corresponding eigenfunctions. Intuitively, the trace of etΔ is defined as

 tr  e−tΔ=∑k=1∞ 〈〈e−tΔ ψk,ψk〉〉 (47)

This is clearly keλkt or μetμdimΓμ(E) , so the definition of tr is well‐defined if and only if

 ∑k e−λkt<∞ (48)

Theorem 3.3. For any p,qM , let {e1(p),,eN(p)} and {f1(q),,fN(q)} be orthonormal bases on Ep and Eq , respectively, then the following two results hold for t>0 ,

 (a)  ∫M∫∑a,b=1N 〈G(t,q,p)ea(p),fb(q)〉2 dvqdvp<∞,(b)  ∑k=1∞ e2λkt<∫M∫∑a,b=1N 〈G(t,q,p)ea(p),fb(q)〉2 dvqdvp<∞ (49)

Proof: When t>0 , G(t,q,p) is continuous and hence satisfies (a). For and wΓ(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|2dvq=k=1e2λktψk(p),w2

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

a=1NM|G(t,q,p)ea(p)|2dvq=a=1Nk=1e2λktψk(p),ea(p)2=k=1a=1Ne2λktψk(p),ea(p)2=k=1e2λktψk(p),ψk(p)

Then for any m , it follows that

k=1me2λkt=k=1mMe2λktψk(p),ψk(p)dvpMk=1e2λktψk(p),ψk(p)dvp=MdvpMa=1N|G(t,q,p)ea(p)|2dvq=MMa,b=1NG(t,q,p)ea(p),fb(q)2dvqdvp<

Theorem 3.4. For any t>0 ,

 tr  (e−tΔ)=∫Mtr G(t,p,p) dvp (50)

Proof: From Theorem 2.2, it follows that

trG(t,p,p)=a=1NG(t,p,p)ea(p),ea(p)=a=1Nk=1etλkψk(p)ea(p)ψk(p),ea(p)=a=1Nk=1etλkψk(p),ea(p)2=k=1etλkψk(p),ψk(p)2

Integrating this on both sides, it is found that

MtrG(t,p,p)dvp=Mk=1etλkψk(p),ψk(p)2dvp=k=1etλk=tr(etΔ)

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

 limt→∞  tr  e−tΔ=∑k=1∞ limt→∞ e−tλk=dim Γ0(E) (51)

In fact, as t0 , the equality

G(t,p,p)=1(4πt)n/2+O(1tn/2)

and the previous theorem imply that limt0tretΔ= .

## 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)=tretD*DtretDD*=MtrG+(t,p,p)dvpMtrG(t,p,p)dvp

by Theorem 3.5, where G±(t,p,p) are the fundamental solutions of t+D*D and t+DD* . As t0 , Theorem 3.2 assumes the form

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

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

 ∑k e−λkt=1(4πt)n/2∑k=0∞ ∫M uk(x,x) dvx (52)

Proof:

keλkt=MtrG(t,x,x)dvx=1(4πt)n/2k(Muk(x,x)dvx)tk

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

ieλit1(4πt)n/2k=0aktk,ak=Muk(x,x)dvx

and the trace does not appear in the case of functions. The superscript on the Laplacian Δp 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 {λi} denote the spectrum of both M and N with dimM=m and dimN=n . Then it follows that

1(4πt)m/2k=0(MukM(p,p)dvp)tk=i=0eλit=1(4πt)n/2k=0(NukN(q,q)dvq)tk

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

1(4πt)m/2[Mu0M(p,p)dvpNuN(q,q)dvq]=1(4πt)m/2k=1(MukM(p,p)dvpNuN(q,q)dvq)tk

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

 ∫M u0M(p,p) dvp=∫N u0N(q,q) dvq (53)

Iterating this argument leads to the set of equations

 ∫MukM(p,p) dvp=∫N ukN(q,q) dvq (54)

for all k>0 . In particular, since u0=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 Mukdvp . The first integral contains basic geometric information. It is then natural to investigate the other integrals in sequence as well. Recall that Rp,Rp, 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(Rp,Rp,,kRp) 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 ,

 u1(p,p)=P1n(Rp),  uk(p,p)=Pkn(Rp,∇Rp,…,∇2k−2Rp),  k≥2 (55)

for some universal polynomials Pkn .

Thus, P1n is a linear function with no constant term and u1(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 u1(p,p) on a manifold is the scalar curvature R(p)=Rijij andso there exists a constant C such that u1(p,p)=CR(p) .

Theorem 4.3.

 u1(p,p)=16R(p) (56)

Proof: The proof amounts to noticing that P1n is a universal polynomial, so it suffices to compute C over one kind of manifold. A good choice is to integrate over Sn with the standard metric and work it out explicitly in normal coordinates. It is found that u1(p,p)=n(n1)/6 andit is known that R(p)=n(n1) for all pSn 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=0eλkt=1(4πt)n/2{v(M)+16MR(x)dvxt+O(t2)}

where v(M) is the volume of M .

Suppose that λ is positive and here we let Eλp denote the possibly trivial eigenspace of Δ on p ‐forms. If ωEλp then it follows that Δp+1dω=dΔpω=λdω , hence dωEλp+1 . Thus, a well‐defined sequential ordering of the spaces can be established. If ωEλp has the property that dω=0 , then λω=Δpω=(δd+dδ)ω=dδω . Therefore, since λ0 , it is found that ω=d(1λδω) . Thus, the sequence 0Eλ0ddEλn0 is exact. Since the operator d+δ is an isomorphism on kEλ2k , it follows that

 ∑s (−1)sdimEλs=0 (57)

Theorem 4.4. Let {λis} be the spectrum of the operator Δ , then

 ∑s(−1)s ∑i e−λist=∑s(−1)sdimkerΔs. (58)

Proof: By (57),

s(1)skeλkst=s(1)seλit

The sum on the right is only over eigenvalues such that λip=0 and so

eλipt=dimkerΔp.

This has the consequence that

 ∑p (−1)p tr e−tΔ=∑p (−1)p∑k e−λkpt (59)

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 tretΔ 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)pdimHdHp(M)=∑p(−1)pdimkerΔp=∑p(−1)p tr e−tΔp=∑p(−1)p ∫M  tr  G(t,x,x) dvx (60)

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

 χ(M)=1(4πt)n/2 ∑k=0∞(∫M ∑s=0n  tr  uks(x,x) dvx)tk (61)

The uks in Eq. (61) are the coefficients in the asymptotic expansion for tr(etΔs) . 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(4π)n/2 ∫M ∑s=0n (−1)s tr  uks(x,x) dvx={0,k≠n2;χ(M),k=n2. (62)

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

 χ(M)=12π∫M K daM (63)

Proof: By the last theorem and the fact that trukp(x,x)=trukp1(x,x) , it follows that

 χ(M)=14π∫M∑p=02 (−1)p tr  u1p daM=14π∫M( tr  u10− tr  u11+ tr  u12) daM=14π∫M (2 tr  u10− tr  u11) daM=14π ∫M (23K− tr  u11) daM (64)

since the scalar curvature is two times the Gaussian. Now it must be that tru11(x,x)=CR(x)=2CK(x) , for some constant C . The standard sphere S2 has Gaussian curvature one andso C can be calculated from Eq. (64),

2=12πS2(13C)daM=12π(13C)(4π)

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 u1 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,

Mu1M(x,x)dvx=Nu1N(y,y)dvy

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

 6πχ(M)=∫M u1M(x,x) dvx=∫N u1N(y,y) dvy=6πχ(N) (65)

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

## 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 132π2{K24|ρr|2+|R|2} , where K is the scalar curvature, |ρr|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 Rijkl be the components of the Riemann curvature tensor relative to an arbitrary local frame field {ei} for the tangent bundle TM and adopt the Einstein summation convention. Let m=2s be even, then the Pfaffian Em(g) is defined to be

 Em(g)=1(8π)ss! Ri1i2j2j1⋯Ri2s−1i2sj2sj2s−1 g(ei1∧⋯∧ei2s,ej1∧⋯∧ej2s) (66)

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 Em(g) dvM (67)

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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