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.
- differential form
- Hodge theory
- partial differential operator
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 ‐forms to 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 ‐forms is perhaps the most well known, as well as other elliptic operators.
On a compact manifold, the spectrum of the Laplacian on ‐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 ‐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 ‐forms with its solution given by the heat semigroup [1–3].
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 [4–6].
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 .
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 ‐dimensional Riemannian manifold , an orthonormal moving frame can be chosen with the accompanying dual coframe which satisfy
It is then possible to define a system of one‐forms and two‐forms by solving the equations,
It then follows that the Christoffel coefficients and components of the Riemann tensor for are
The inner product induced by the Riemannian metric on is denoted here by and it induces a metric on as well. Using the Riemannian metric and the measure on , an inner product denoted can be defined on so that for ,
where if is a system of local coordinates,
is the Riemannian measure on . Clearly, is linear with respect to , andwith equality if and only if . Hodge introduced a star homomorphism * , which is defined next.
where is , , or depending on whether is an even or odd permutation of , respectively.
(ii) If is an oriented Riemannian manifold with dimension , define the operator
In terms of the two operators and , the Laplacian acting on ‐forms can be defined on the two subspaces
The operator can be regarded as the operators on these subspaces,
is called the Hodge‐de Rham operator. It has the property that it is a self‐conjugate operator, and . It is useful in studying the Laplacian to have a formula for the operator and hence for and as well.
Let be an orthonormal moving frame defined on an open set . Define as well the pair of operators
If is a Riemannian manifold and is a Levi‐Civita connection, then a connection on the space , namely , can also be defined such that
The connection may be regarded as a first‐order derivative operator .
In terms of the operator (Eq. (12)), define a second‐order differential operator by
where is an orthonormal moving frame. The operator in Eq. (13) is referred to as the
In Eq. (14), is the scalar curvature, and 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 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.
with parameter and satisfies the following three conditions, the family is called a fundamental solution of the heat operator (15) where . First, is an
where acts on the variable . Finally, if is a continuous section of the vector bundle , then
for all , where is the volume measure with respect to the coordinates of given in terms of the Riemannian metric.
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 is a self‐adjoint nonnegative Schrödinger operator, then there exists a set of sections such that
Moreover, denoting the completion of the inner product space by , the set is a complete set in , so for any ,
Finally, the set satisfies the equation
where are the eigenvalues of andform an increasing sequence: where .
Denote by when and satisfies the semigroup property and is a self‐adjoint, compact operator.
with holds in .
Differentiating with respect to and using , we get
It follows from this that
and since depend linearly on , so , where is a linear function. There exists independent of such that so that
Consequently, for any , we have
Moreover, can also be expanded in terms of the basis set,
Upon comparing these last two expressions, it is clear that for all andwe are done.
One application of the heat equation method developed so far is to develop and give a proof of the Hodge theorem.
is a finite‐dimensional vector space.
For any , there is a unique decomposition of as , where and .
The first part is a direct consequence of the expansion theorem and due to the fact , 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 . Define
Since , it follows that andthe ‐th de Rham cohomology group of is defined to be
From Eq. (21), construct
In 1935, Hodge claimed a theorem, which stated every element in can be represented by a unique harmonic form , one which satisfies both and . Denote the set of harmonic forms as .
To finish the proof, it suffices to show that . If , that is , then
This implies that and , hence .
is a finite dimensional vector space for .
There is an orthogonal decomposition of as
The ‐th Betti number defined as also satisfies for .
3. The Minakshisundaran‐Pleijel paramatrix
Let be a Riemannian manifold with dimension and a vector bundle over with an inner product and a metric connection. Here, the following formal power series is considered with a special transcendental multiplier and parameters , defined by
In Eq. (26), the function is the metric distance between and in , is the fiber of over and are ‐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 is chosen in a neighborhood of point and is centered at . This means that if is in this neighborhood about , which has coordinates , then the function is
In terms of these coordinates, we calculate the components of ,
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 is calculated
It is very convenient to abbreviate the function appearing in front of the sum in Eq. (30) as follows:
Let be a frame that is parallel along geodesics passing through and satisfies
The individual components of (32) can be calculated as follows; since is a function and so
and the Laplace‐Beltrami operator on the function is given by
Substituting these results into (32), it follows that
This is summarized in the following Lemma.
for all and Eq. (38) is initialized with .
In fact, for fixed and , there always exists a unique solution to problem (Eq. (38)) over a small coordinate neighborhood about .
and is with respect to and . Since is , hence . Consider integrating against ,
The integral of Eq. (42) over can be broken up into an integral over anda second integral over the set . On the latter set, the limit converges uniformly hence
To estimate the remaining integral, choose a normal coordinate system at and denote the integration coordinates as , then the integrand of Eq. (42) is given as
Therefore, in the limit using Definition 2.4,
This result implies that
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.
and the symbol on the right‐hand side of Eq. (44) signifies a quantity with the property that
From the previous theorem and existence and regularity of the fundamental solution, the result of Levi iteration initialized by is exactly the fundamental solution. Equality (Eq. (41)) means that there exists a constant such that for any ,
Let be the volume of the manifold . Using this result, the following upper bound is obtained
We have set . Exactly the same procedure applies to . Based on the pattern established this way, induction implies that the following bound results
The formula for Levi iteration yields upon summing this over the following upper bound
Using this bound, the required estimate is obtained,
This finishes the proof.
Now if all the Hodge theorem is used, formal expressions for the index can be obtained. Suppose is an operator such that and are Schrödinger operators and is the adjoint of . Suppose the operators and are defined, so they are self‐adjoint and have nonnegative real eigenvalues. Then the spaces and can be defined this way
For any , the dimensions of the spaces in (44) are finite and moreover,
Consequently, an expression for the index can be obtained from Eq. (45) as follows
where is the fundamental solution of heat operator (Eq. (15)).
Let be the eigenvalues of the operator and the corresponding eigenfunctions. Intuitively, the trace of is defined as
This is clearly or , so the definition of is well‐defined if and only if
Replacing by the basis element , this implies that
Then for any , it follows that
Integrating this on both sides, it is found that
Note that Eq. (48) is a series with positive terms which converges uniformly as . Therefore,
In fact, as , the equality
and the previous theorem imply that .
4. An application of the expansions: the Gauss Bonnet theorem
As far as is concerned, it is the case for all that,
by Theorem 3.5, where are the fundamental solutions of and . As , Theorem 3.2 assumes the form
The spectrum of the Laplacian on functions characterizes a lot of interesting geometric information. Note that Eq. (52) can be written as
and the trace does not appear in the case of functions. The superscript on the Laplacian 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.
This implies that , which in turn implies that
Since the right‐hand side of the equation depends on , but the left‐hand side does not, this result implies that
Iterating this argument leads to the set of equations
for all . In particular, since , Eq. (53) leads to the conclusion .
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 . The first integral contains basic geometric information. It is then natural to investigate the other integrals in sequence as well. Recall that denote the covariant derivatives of the curvature tensor at . A polynomial in the curvature and its covariant derivatives is called universal if its coefficients depend only on the dimension of . The notation 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 . The following theorem will not be proved, but it will be used shortly.
for some universal polynomials .
Thus, is a linear function with no constant term and is a linear function of the components of the curvature tensor at , with no covariant derivative terms. The only linear combination of curvature components that produces a well‐defined function on a manifold is the scalar curvature andso there exists a constant such that .
The large 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 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 is the volume of .
Suppose that is positive and here we let denote the possibly trivial eigenspace of on ‐forms. If then it follows that , hence . Thus, a well‐defined sequential ordering of the spaces can be established. If has the property that , then . Therefore, since , it is found that . Thus, the sequence is exact. Since the operator is an isomorphism on , it follows that
The sum on the right is only over eigenvalues such that and so
This has the consequence that
is independent of the parameter . This means that its large or long behavior is the same as its short or small behavior. To put it another way, the long‐time behavior of 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
From the asymptotic expansion theorem, the following expression for results
The in Eq. (61) are the coefficients in the asymptotic expansion for . Since is independent of , only the constant or ‐independent term on the right‐hand side of Eq. (61) can be nonzero. This implies the following important theorem.
since the scalar curvature is two times the Gaussian. Now it must be that , for some constant . The standard sphere has Gaussian curvature one andso can be calculated from Eq. (64),
As an application of this theorem, note that the calculation of gives another topological obstruction to manifolds having the same spectrum.
On a surface, the scalar curvature is twice the Gaussian curvature, so by the Gauss‐Bonnet theorem,
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 , where is the scalar curvature, is the norm of the Ricci tensor, 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 be the components of the Riemann curvature tensor relative to an arbitrary local frame field for the tangent bundle and adopt the Einstein summation convention. Let be even, then the Pfaffian is defined to be
The Euler characteristic 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 , 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.