ScholarWorks @ UTRGV ScholarWorks @ UTRGV Spectral Theory of Operators on Manifolds Spectral Theory of Operators on Manifolds

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 signifi-cant 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 Minakshi-sundaram-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.


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 þ 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 [1][2][3].
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 [4][5][6].
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].

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 It is then possible to define a system of one-forms ω ij and two-forms Ω ij by solving the equations, It then follows that the Christoffel coefficients and components of the Riemann tensor for M are The inner product induced by the Riemannian metric on M is denoted here by 〈 Á ; Á 〉 : ΓðTMÞ · ΓðTMÞ!F ðMÞ and it induces a metric on Λ k ðMÞ as well. Using the Riemannian metric and the measure on M, an inner product denoted 〈〈 Á ; Á 〉〉 : Λ k ðMÞ · Λ k ðMÞ!R can be defined on Λ k ðMÞ so that for α; β ∈ Λ k ðMÞ, where if ðx 1 ;…;x m Þ is a system of local coordinates, 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 ¼ 0. Hodge introduced a star homomorphism * : Λ k ðMÞ! Λ n−k ðMÞ, which is defined next.
(ii) If M is an oriented Riemannian manifold with dimension n, define the operator In terms of the two operators d and δ, the Laplacian acting on k-forms can be defined on the two subspaces Λ even ðMÞ¼⊕ even Λ k ðMÞ; The operator d þ δ can be regarded as the operators on these subspaces, Definition 2.2. Let M be a Riemannian manifold, then the operator is called the Hodge-de Rham operator. It has the property that it is a self-conjugate operator, It is useful in studying the Laplacian to have a formula for the operator Δ ¼ðd þ δÞ 2 and hence for D Ã 0 D 0 and D Ã 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 Lemma 2.1. The operators E AE j satisfy the following relations If M is a Riemannian manifold and ∇ : ΓðTMÞ · ΓðTMÞ!ΓðTMÞ is a Levi-Civita connection, then a connection on the space Λ Ã ðMÞ, namely ðX;ωÞ!∇ X ω, can also be defined such that ð∇ X ωÞðYÞ¼XðωðYÞÞ−ωð∇ X YÞ; Y ∈ ΓðTMÞ The connection may be regarded as a first-order derivative operator ðX;Y;ωÞ!DðX;YÞω.
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 ¼ −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 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. Gðt;q;pÞ : 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 ¼ π −1 ðpÞ. First, Gðt;q;pÞ : 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 θðt;qÞ¼Gðt;q;pÞw, 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 Hθðt;qÞ¼0, which can be written as where Δ q acts on the variable q. Finally, if ϕ is a continuous section of the vector bundle E, then lim t!0 þ ð M Gðt;q;pÞϕðpÞ dv p ¼ ϕðqÞ for all ϕ, where dv 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 ðt;q;pÞ is given. The following procedure taking G 0 ðt;q;pÞ to Gðt;q;pÞ is called the Levi algorithm: Gðt; q; pÞ¼G 0 ðt; q; pÞþ 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 : ΓðEÞ!ΓðEÞ is a self-adjoint nonnegative Schrödinger operator, then there exists a set of C ∞ sections {ψ i }⊂ ΓðEÞ such that Moreover, denoting the completion of the inner product space ΓðEÞ by ΓðEÞ, the set {ψ i }i sa complete set in ΓðEÞ, so for any ψ ∈ ΓðEÞ, Finally, the set {ψ i } satisfies the equation where λ i are the eigenvalues of Δ andform an increasing sequence: Denote Uðt;qÞ by ðT t ψÞðqÞ when Uð0;qÞ¼ψðqÞ and T t satisfies the semigroup property andT t is a self-adjoint, compact operator.
Theorem 2.2. Let Gðt; q; pÞ be the fundamental solution of the heat operator (15), then with w ∈ E p holds in ΓðEÞ.
Proof: For fixed t > 0 and w∈E p , expand Gðt; q; pÞw in terms of eigenfunctions ψ i ðqÞ, Differentiating with respect to t and using It follows from this that There existsc i ðpÞ independent of w such that c i ðpÞw ¼ 〈c i ðpÞ;w〉 so that Consequently, for any β ∈ ΓðEÞ, we have One application of the heat equation method developed so far is to develop and give a proof of the Hodge theorem.
The first part is a direct consequence of the expansion theorem and due to the fact H⊥ΔðΓðEÞÞ, 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 Ã dR ðMÞ. Define In 1935, Hodge claimed a theorem, which stated every element in H k dR ðMÞ can be represented by a unique harmonic form α, one which satisfies both dα ¼ 0 and δα ¼ 0. Denote the set of harmonic forms as H k ðMÞ.
Proof: Since Δ ¼ dδ þ δd, this implies that ΔðΛ k ðMÞÞ ⊂ Λ k ðMÞ andit is clear that To finish the proof, it suffices to show that ker{Δ : The k-th Betti number defined as b k ðMÞ¼dimH k ðM; RÞ also satisfies b k ðMÞ¼b n−k ðMÞ for 0 ≤ k ≤ n.

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 =4t and parameters ðt;p;qÞ∈ð0;∞Þ · M · M, defined by In Eq. (26), the function ρ ¼ ρðp;qÞ is the metric distance between p and q in M, E p ¼ π −1 ðpÞ is the fiber of E over p and u k ðp;qÞ : 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: 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 ðx 1 ;…;x n Þ, then the function ρðp;qÞ is In terms of these coordinates, we calculate the components of g, and define the differential operator 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 It is very convenient to abbreviate the function appearing in front of the sum in Eq. (30) as follows: ΦðρÞ¼ Let {e 1 ;…;e n } be a frame that is parallel along geodesics passing through p and satisfies In terms of the function in Eq. (31), the operator Δ 0 acting on Eq. (26) is given as The individual components of (32) can be calculated as follows; since Φ is a function ∇ e a Φ ¼ e a Φ and so e a ΦðρÞ¼Φ ′ ðρÞe a ðρÞ; Consequently, and the Laplace-Beltrami operator on the function Φ is given by 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, Substituting these results into (32), it follows that Combining Eq. (36) with the derivative of H ∞ with respect to t in Eq. (35), the following version of the heat equation results: This is summarized in the following Lemma.
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. Then G 0 ðt;q;pÞ is a k-th initial solution of the heat operator (15), where k ¼ ⌊ 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 and u N ðp;qÞ is C ∞ with respect to p and q. Since t N− n 2 e −ρ 2 =4t is C k ð½0;∞Þ · M · MÞ, hence Hðϕðp;qÞH N ðt;q;pÞÞ∈C k ð½0;∞Þ · M · MÞ. Consider integrating G 0 against ψðs;βÞ, 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.
Let vðMÞ be the volume of the manifold M. Using this result, the following upper bound is obtained We have set B ¼ A Á T N− n 2 vðMÞ. Exactly the same procedure applies to jK 2 ðt;q;pÞj. Based on the pattern established this way, induction implies that the following bound results The formula for Levi iteration yields upon summing this over m the following upper bound jKðt;q;pÞj ≤ ∑ ∞ m¼0 jK m ðt;q;pÞj ≤ A Á e Bt t N− n 2 Using this bound, the required estimate is obtained, jGðt;q;pÞ−G 0 ðt;q;pÞj ≤ j This finishes the proof.
Now if all the Hodge theorem is used, formal expressions for the index can be obtained. Suppose D : ΓðEÞ!ΓðFÞ is an operator such that D Ã D and DD Ã are Schrödinger operators andD Ã is the adjoint of D. Suppose the operators D Ã D : ΓðEÞ!ΓðEÞ and DD Ã : ΓðFÞ!ΓðEÞ are defined, so they are self-adjoint and have nonnegative real eigenvalues. Then the spaces Γ μ ðEÞ and Γ μ ðFÞ can be defined this way For any m > 0, the dimensions of the spaces in (44) are finite and moreover, Γ 0 ðEÞ¼ker{D : ΓðEÞ!ΓðFÞ}; Γ 0 ðFÞ¼ker{D Ã : ΓðFÞ!ΓðEÞ} Consequently, an expression for the index Ind ðDÞ can be obtained from Eq. (45) as follows where Gðt;q;pÞ 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 −tΔ is defined as This is clearly ∑ k e −λ k t or ∑ μ e −tμ dim Γ μ ðEÞ, so the definition of tr is well-defined if and only if Note that Eq. (48) is a series with positive terms which converges uniformly as t ! ∞. Therefore, In fact, as t ! 0, the equality Gðt;p;pÞ¼ 1 and the previous theorem imply that lim t!0 tr e −tΔ ¼ ∞.
4. An application of the expansions: the Gauss Bonnet theorem   for all k > 0. In particular, since u 0 ¼ 1, Eq. (53) leads to the conclusion vol ðMÞ¼ vol ðNÞ.
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 ð M u k dv 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 PðR p ;∇R p ;…;∇ k R p Þ 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. for some universal polynomials P n k .
Thus, P n 1 is a linear function with no constant term and u 1 ðp;pÞ 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 ðp;pÞ on a manifold is the scalar curvature RðpÞ¼R ij ij andso there exists a constant C such that u 1 ðp;pÞ¼C Á RðpÞ.  Proof: The proof amounts to noticing that P n 1 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 ðp;pÞ¼nðn−1Þ=6 andit is known that RðpÞ¼nðn−1Þ 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 where vðMÞ is the volume of M. since the scalar curvature is two times the Gaussian. Now it must be that tr u 1 1 ðx;xÞ¼ CRðxÞ¼2CKðxÞ, for some constant C. The standard sphere S 2 has Gaussian curvature one andso C can be calculated from Eq. (64), Therefore, C ¼ −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. However, oriented surfaces with the same Euler characteristic are diffeomorphic.

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 32π 2 {K 2 −4jρ r j 2 þjRj 2 }, where K is the scalar curvature, jρ r j 2 is the norm of the Ricci tensor, jRj 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 ijkl be the components of the Riemann curvature tensor relative to an arbitrary local frame field {e i } for the tangent bundle TM and adopt the Einstein summation convention. Let m ¼ 2s be even, then the Pfaffian E m ðgÞ is defined to be E m ðgÞ¼ 1 ð8πÞ s s! R i 1 i 2 j 2 j 1 ⋯R i 2s−1 i 2s j 2s j 2s−1 gðe i 1 ∧⋯∧e i 2s ;e j 1 ∧⋯∧e j 2s Þ (66) The Euler characteristic χðMÞ of any compact manifold of odd dimension without boundary vanishes. Only the even dimensional case is of interest. 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.