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

$$
\omega\_i(e\_j) = \delta\_{ij}, \qquad i, j = 1, \ldots, n \tag{1}
$$

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

$$\nabla\_X \varepsilon\_i = \sum\_j \omega\_{ji}(X)\,\varepsilon\_j, \qquad R(X,Y)\varepsilon\_i = \sum\_j \Omega\_{ji}(X,Y)\varepsilon\_j \tag{2}$$

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

$$
\langle \omega\_{\vec{\mu}} (e\_k) = \sum\_a \langle \omega\_{a\vec{\}}(e\_k) e\_a, e\_i \rangle\_{\mathcal{g}} = \langle \nabla\_{e\_k} e\_{\vec{\mu}}, e\_i \rangle\_{\mathcal{g}} = \Gamma\_{k\vec{\mu}}^i \tag{3}
$$

$$\left< \Omega\_{i\rangle}(e\_k, e\_s) = \sum\_a \left< \Omega\_{a\rangle}(e\_k, e\_s) e\_a, e\_i \right>\_{\mathcal{S}} = \left< \left< R(e\_k, e\_s) e\_j, e\_i \right>\_{\mathcal{S}} = R\_{k\bar{\eta}i} \tag{4}$$

The inner product induced by the Riemannian metric on M is denoted here by 〈 � ; � 〉 : ΓðTMÞ · <sup>Γ</sup>ðTMÞ ! <sup>F</sup>ðM<sup>Þ</sup> and it induces a metric on <sup>Λ</sup><sup>k</sup> ðMÞ as well. Using the Riemannian metric and the measure on <sup>M</sup>, an inner product denoted 〈〈 � ; � 〉〉 : <sup>Λ</sup><sup>k</sup> <sup>ð</sup>M<sup>Þ</sup> · <sup>Λ</sup><sup>k</sup> ðMÞ ! R can be defined on Λk <sup>ð</sup>M<sup>Þ</sup> so that for <sup>α</sup>; <sup>β</sup><sup>∈</sup> <sup>Λ</sup><sup>k</sup> ðMÞ,

$$
\langle\langle\alpha,\beta\rangle\rangle = \int\_M \langle\alpha,\beta\rangle\_{\mathcal{S}} \, dv\_M \tag{5}
$$

where if <sup>ð</sup>x<sup>1</sup>;…;x<sup>m</sup><sup>Þ</sup> is a system of local coordinates,

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

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

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

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

respect to the geometry of the manifold [4–6].

68 Manifolds - Current Research Areas

such as to gauge theories and gravity [7].

greater than two [16, 17].

2. Geometrical preliminaries

$$dv\_M = \det(\mathcal{g}\_{i\hat{\jmath}}) \, d\mathfrak{x}^1 \wedge ... \wedge d\mathfrak{x}^m$$

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 <sup>α</sup> <sup>¼</sup> 0. Hodge introduced a star homomorphism \* : <sup>Λ</sup><sup>k</sup> ðMÞ ! Λ<sup>n</sup>−<sup>k</sup> ðMÞ, which is defined next.

Definition 2.1. (i) For ω ¼ ∑<sup>i</sup>1<⋯<ik f <sup>i</sup>1⋯ik ω<sup>i</sup><sup>1</sup> ∧⋯ωik , define

$$\begin{array}{c} \iota^\*\omega = \sum\_{\substack{\dot{i}\_1 < \cdots < \dot{i}\_k \\ \dot{j}\_1 < \cdots < \dot{j}\_{n-k}}} f\_{\dot{i}\_1 \cdots \dot{i}\_k} \epsilon(\dot{i}\_1, \dots, \dot{i}\_k, \dot{j}\_1, \dots, \dot{j}\_{n-k}) \omega\_{\dot{j}\_1} \wedge \cdots \wedge \omega\_{\dot{\omega}\_{n-k}}, \\ \dot{j}\_1 < \cdots < \dot{j}\_{n-k} \end{array}$$

where E is 1, −1, or 0 depending on whether ði1;…;ik;j <sup>1</sup>;…;j <sup>n</sup>−kÞ is an even or odd permutation of ð1;…;nÞ, respectively.

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

$$\delta = (-1)^{nk+n+1\*} d^\* : \Lambda^k(M) \to \Lambda^{k-1}(M) \tag{6}$$

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

$$
\Lambda^{\text{even}}(M) = \oplus\_{\text{even}} \Lambda^k(M), \qquad \Lambda^{\text{odd}}(M) = \oplus\_{\text{odd}} \Lambda^k(M) \tag{7}
$$

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

$$D\_0 = d + \delta : \Lambda^{\text{even}}(M) \to \Lambda^{\text{odd}}(M), \quad D\_1 = d + \delta : \Lambda^{\text{odd}}(M) \to \Lambda^{\text{even}}(M) \tag{8}$$

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

$$D\_0 = d + \delta : \Lambda^{\text{even}}(M) \to \Lambda^{\text{odd}}(M) \tag{9}$$

is called the Hodge-de Rham operator. It has the property that it is a self-conjugate operator, D� <sup>0</sup> ¼ D<sup>1</sup> and D� <sup>1</sup> ¼ D0. It is useful in studying the Laplacian to have a formula for the operator Δ ¼ ðd þ δÞ <sup>2</sup> and hence for D� <sup>0</sup>D<sup>0</sup> and D� <sup>1</sup>D<sup>1</sup> as well.

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

$$E\_j^+ = \omega\_{\not\!\!/} \wedge \cdot + i(\mathfrak{e}\_{\not\!\!/}) : \Lambda^\*(\mathcal{U}) \to \Lambda^\*(\mathcal{U}), \qquad E\_{\not\!\!/} = \omega\_{\not\!\!/} \wedge \cdot - i(\mathfrak{e}\_{\not\!\!/}) : \Lambda^\*(\mathcal{U}) \to \Lambda^\*(\mathcal{U}) \tag{10}$$

Lemma 2.1. The operators E� <sup>j</sup> satisfy the following relations

$$E\_i^+ E\_j^+ + E\_j^+ E\_i^+ = 2\delta\_{i\bar{\jmath}}, \quad E\_i^+ E\_{\bar{\jmath}}^- + E\_{\bar{\jmath}}^- E\_i^+ = 0, \quad E\_i^- E\_{\bar{\jmath}}^- + E\_{\bar{\jmath}}^- E\_i^- = -2\delta\_{i\bar{\jmath}} \tag{11}$$

If M is a Riemannian manifold and ∇ : ΓðTMÞ · ΓðTMÞ ! ΓðTMÞ is a Levi-Civita connection, then a connection on the space Λ� ðMÞ, namely ðX;ωÞ ! ∇Xω, can also be defined such that

$$(\nabla\_X \omega)(Y) = X(\omega(Y)) \neg \omega(\nabla\_X Y), \quad Y \in \Gamma(TM)$$

The connection may be regarded as a first-order derivative operator ðX;Y;ωÞ ! DðX;YÞω.

Definition 2.3. The second-order derivative operator ðX;Y;ωÞ ! DðX;YÞω is defined to be

$$D(X,Y)\omega = \nabla\_X \nabla\_Y \omega \! - \nabla\_{\nabla\_X Y} \omega \! \tag{12}$$

In terms of the operator (Eq. (12)), define a second-order differential operator Δ<sup>0</sup> : Λ� ðMÞ ! Λ� ðMÞ by

$$
\Delta\_0 = \sum\_i D(e\_i, e\_i),
\tag{13}
$$

where {ei} n <sup>1</sup> is an orthonormal moving frame. The operator Δ<sup>0</sup> 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} n <sup>1</sup> . The Laplace operator can be expressed as

Spectral Theory of Operators on Manifolds http://dx.doi.org/10.5772/64611 71

$$
\Delta = \left(d + \delta\right)^2 = -\Delta\_0 - \frac{1}{8} \sum\_{i,j,k,s} R\_{ijks} \, E\_i^+ E\_j^+ E\_k^- E\_s^- + \frac{1}{4} R \tag{14}
$$

In Eq. (14), R is the scalar curvature, R ¼ −∑<sup>i</sup>;jRijij and Δ<sup>0</sup> is the Laplace-Beltrami operator (13).

<sup>Λ</sup>evenðMÞ ¼ <sup>⊕</sup>even <sup>Λ</sup><sup>k</sup>

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

<sup>0</sup>D<sup>0</sup> and D�

ðUÞ ! Λ�

<sup>i</sup> ¼ 2δij; E<sup>þ</sup>

D�

Λ�

ðMÞ by

where {ei}

n

Laplace-Beltrami operator.

n

mal moving frame {ei}

<sup>0</sup> ¼ D<sup>1</sup> and D�

70 Manifolds - Current Research Areas

pair of operators

E<sup>þ</sup>

Lemma 2.1. The operators E�

E<sup>þ</sup> <sup>i</sup> E<sup>þ</sup> <sup>j</sup> þ E<sup>þ</sup> <sup>j</sup> E<sup>þ</sup>

then a connection on the space Λ�

<sup>2</sup> and hence for D�

<sup>j</sup> ¼ ω<sup>j</sup> ∧ � þiðejÞ : Λ�

Δ ¼ ðd þ δÞ

<sup>ð</sup>MÞ; <sup>Λ</sup>oddðMÞ ¼ <sup>⊕</sup>odd <sup>Λ</sup><sup>k</sup>

<sup>D</sup><sup>0</sup> <sup>¼</sup> <sup>d</sup> <sup>þ</sup> <sup>δ</sup> : <sup>Λ</sup>evenðMÞ ! <sup>Λ</sup>oddðMÞ; <sup>D</sup><sup>1</sup> <sup>¼</sup> <sup>d</sup> <sup>þ</sup> <sup>δ</sup> : <sup>Λ</sup>oddðMÞ ! <sup>Λ</sup>evenðM<sup>Þ</sup> (8)

<sup>1</sup> ¼ D0. It is useful in studying the Laplacian to have a formula for the operator

<sup>i</sup> <sup>¼</sup> <sup>0</sup>; <sup>E</sup><sup>−</sup>

is called the Hodge-de Rham operator. It has the property that it is a self-conjugate operator,

Let {e1;…;en} be an orthonormal moving frame defined on an open set U. Define as well the

<sup>ð</sup>UÞ; <sup>E</sup><sup>−</sup>

If M is a Riemannian manifold and ∇ : ΓðTMÞ · ΓðTMÞ ! ΓðTMÞ is a Levi-Civita connection,

ð∇XωÞðYÞ ¼ XðωðYÞÞ−ωð∇XYÞ; Y ∈ ΓðTMÞ

The connection may be regarded as a first-order derivative operator ðX;Y;ωÞ ! DðX;YÞω. Definition 2.3. The second-order derivative operator ðX;Y;ωÞ ! DðX;YÞω is defined to be

In terms of the operator (Eq. (12)), define a second-order differential operator Δ<sup>0</sup> : Λ�

Δ<sup>0</sup> ¼ ∑ i

<sup>1</sup> is an orthonormal moving frame. The operator Δ<sup>0</sup> in Eq. (13) is referred to as the

Theorem 2.1. (Weitzenböck) Let M be a Riemannian manifold M with an associated orthonor-

<sup>1</sup> . The Laplace operator can be expressed as

<sup>j</sup> satisfy the following relations

<sup>i</sup> <sup>E</sup><sup>−</sup> <sup>j</sup> <sup>þ</sup> <sup>E</sup><sup>−</sup> <sup>j</sup> E<sup>þ</sup>

<sup>1</sup>D<sup>1</sup> as well.

<sup>D</sup><sup>0</sup> <sup>¼</sup> <sup>d</sup> <sup>þ</sup> <sup>δ</sup> : <sup>Λ</sup>evenðMÞ ! <sup>Λ</sup>oddðM<sup>Þ</sup> (9)

<sup>j</sup> ¼ ω<sup>j</sup> ∧ � −iðejÞ : Λ�

i E− <sup>j</sup> <sup>þ</sup> <sup>E</sup><sup>−</sup> j E−

ðMÞ, namely ðX;ωÞ ! ∇Xω, can also be defined such that

DðX;YÞω ¼ ∇X∇Yω−∇∇XYω (12)

Dðei;eiÞ; (13)

ðUÞ ! Λ�

ðUÞ (10)

ðMÞ !

<sup>i</sup> ¼ −2δij (11)

ðMÞ (7)

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 −Δ0−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

$$\mathcal{H} = \frac{\partial}{\partial t} + \Delta \tag{15}$$

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 : E ! M is a vector bundle with connection. Let Δ<sup>0</sup> : ΓðEÞ ! ΓðEÞ 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 : ΓðEÞ ! ΓðEÞ be a FðMÞ-linear map. Then, Δ ¼ −Δ0−F is a Schrödinger operator. If a family of R-linear maps

$$G(t, q, p) : E\_p \to E\_q$$

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 Ep <sup>¼</sup> <sup>π</sup><sup>−</sup><sup>1</sup>ðpÞ. First, <sup>G</sup>ðt;q;p<sup>Þ</sup> : Ep ! Eq is an R-linear map of vector spaces and continuous in all variables t;q;p. Second, for a fixed w ∈ Ep, let θðt;qÞ ¼ Gðt;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;qÞ ¼ 0, which can be written as

$$G\left(\frac{\partial}{\partial t} + \Delta\_q\right) G(t, q, p) = 0\tag{16}$$

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

$$\lim\_{t \to 0^+} \int\_M G(t, q, p) \wp(p) \, d\upsilon\_p = \wp(q).$$

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 G0ðt;q;pÞ is given. The following procedure taking G0ðt;q;pÞ to Gðt;q;pÞ is called the Levi algorithm:

$$\begin{aligned} K\_0(t, q, p) &= \left(\frac{\partial}{\partial t} + \triangle\_q\right) G(t, q, p), \\ K\_{m+1}(t, q, p) &= \int\_0^t d\tau \int\_M K\_0(t - \tau, q, z) K\_m(\tau, z, p) \, d\upsilon\_z \\ \overline{K}(t, q, p) &= \sum\_{m=0}^\infty (-1)^{m+1} K\_m(t, q, p), \end{aligned} \tag{17}$$
 
$$\begin{aligned} \mathcal{G}(t, q, p) &= \mathcal{G}\_0(t, q, p) + \int\_0^t d\tau \int\_M \mathcal{G}\_0(t - \tau, q, z) \overline{K}(\tau, z, p) \, d\upsilon\_z \end{aligned}$$

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 <sup>C</sup><sup>∞</sup> sections {ψ<sup>i</sup> }⊂ΓðEÞ such that

$$\langle\langle\psi\_i,\psi\_j\rangle\rangle = \int\_M \langle\psi\_i(\mathbf{x}),\psi\_j(\mathbf{x})\rangle \,d\upsilon\_\mathbf{x} = \delta\_{ij}$$

Moreover, denoting the completion of the inner product space ΓðEÞ by ΓðEÞ, the set {ψ<sup>i</sup> } is a complete set in ΓðEÞ, so for any ψ∈ΓðEÞ,

$$\boldsymbol{\psi} = \sum\_{i=1}^{\infty} \left< \langle \boldsymbol{\psi}, \psi\_i \rangle \right> \boldsymbol{\psi}\_i$$

Finally, the set {ψ<sup>i</sup> } satisfies the equation

$$
\Delta \psi\_i = \lambda\_i \psi\_i, \qquad T\_t \psi\_i = e^{-t\lambda\_i} \psi\_i
$$

where λ<sup>i</sup> are the eigenvalues of Δ andform an increasing sequence: 0 ≤ λ<sup>1</sup> ≤ λ<sup>2</sup> ≤ ⋯ where lim<sup>k</sup>!<sup>∞</sup> λ<sup>k</sup> ¼ ∞.

Denote Uðt;qÞ by ðTtψÞðqÞ when Uð0;qÞ ¼ ψðqÞ and Tt satisfies the semigroup property andTt is a self-adjoint, compact operator.

Theorem 2.2. Let Gðt; q; pÞ be the fundamental solution of the heat operator (15), then

$$\mathcal{G}(t, q, p)w = \sum\_{i=1}^{\infty} e^{\lambda\_i t} \langle \,\psi\_i(p), w \rangle \psi\_i(q) \tag{18}$$

with w ∈ Ep holds in ΓðEÞ.

Proof: For fixed t > 0 and w∈Ep, expand Gðt; q; pÞw in terms of eigenfunctions ψ<sup>i</sup> ðqÞ,

#### Spectral Theory of Operators on Manifolds http://dx.doi.org/10.5772/64611 73

$$G(t,q,p)w = \sum\_{i=1}^{\circ \circ} \sigma\_i(t,p,w)\psi\_i(q), \quad \sigma\_i(t,p,w) = \int\_M \langle \psi\_i(q), G(t,q,p)w \rangle \, dv\_q$$

Differentiating with respect to t and using Δψ<sup>i</sup> ¼ λiψ<sup>i</sup> , we get

$$\begin{split} \frac{\partial}{\partial t} \sigma\_i(t, p, w) &= \int\_M \left< \psi\_i(q), \frac{\partial}{\partial t} G(t, q, p) w \right> d\upsilon\_q = \int\_M \left< \psi\_i(q), -\Delta\_q G(t, q, p) w \right> d\upsilon\_q \\ &= -\int\_M \left< \Delta\_q \psi\_i(q), G(t, q, p) w \right> d\upsilon\_q = -\lambda\_i \int\_M \left< \psi\_i(q), G(t, q, p) w \right> d\upsilon\_q \\ &= -\lambda\_i \sigma\_i(t, p, w) \end{split}$$

It follows from this that

<sup>K</sup>0ðt; <sup>q</sup>; <sup>p</sup>Þ ¼ <sup>∂</sup>

ðt 0 dτ ð M

Kðt; q; pÞ ¼ ∑

develop a Fourier expansion theorem applicable to this Schrödinger operator.

Kmþ<sup>1</sup>ðt; q; pÞ ¼

Gðt; q; pÞ ¼ G0ðt; q; pÞ þ

}⊂ΓðEÞ such that

〈〈ψ<sup>i</sup> ;ψj 〉〉 ¼ ð M 〈ψ<sup>i</sup> ðxÞ;ψ<sup>j</sup>

} satisfies the equation

complete set in ΓðEÞ, so for any ψ∈ΓðEÞ,

set of <sup>C</sup><sup>∞</sup> sections {ψ<sup>i</sup>

72 Manifolds - Current Research Areas

Finally, the set {ψ<sup>i</sup>

lim<sup>k</sup>!<sup>∞</sup> λ<sup>k</sup> ¼ ∞.

a self-adjoint, compact operator.

with w ∈ Ep holds in ΓðEÞ.

∂t þ Δ<sup>q</sup> � �

∞ m¼0 ð−1Þ mþ1

ðt 0 dτ ð M

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

Suppose Δ : ΓðEÞ ! ΓðEÞ is a self-adjoint nonnegative Schrödinger operator, then there exists a

Moreover, denoting the completion of the inner product space ΓðEÞ by ΓðEÞ, the set {ψ<sup>i</sup>

〈〈ψ;ψ<sup>i</sup>

; Ttψ<sup>i</sup> ¼ e

where λ<sup>i</sup> are the eigenvalues of Δ andform an increasing sequence: 0 ≤ λ<sup>1</sup> ≤ λ<sup>2</sup> ≤ ⋯ where

Denote Uðt;qÞ by ðTtψÞðqÞ when Uð0;qÞ ¼ ψðqÞ and Tt satisfies the semigroup property andTt is

Theorem 2.2. Let Gðt; q; pÞ be the fundamental solution of the heat operator (15), then

∞ i¼1 e λit 〈 ψ<sup>i</sup>

〉〉 ψ<sup>i</sup>

−tλ<sup>i</sup> ψi

ðpÞ;w〉ψ<sup>i</sup>

ðqÞ (18)

ðqÞ,

ψ ¼ ∑ ∞ i¼1

Δψ<sup>i</sup> ¼ λiψ<sup>i</sup>

Gðt;q;pÞw ¼ ∑

Proof: For fixed t > 0 and w∈Ep, expand Gðt; q; pÞw in terms of eigenfunctions ψ<sup>i</sup>

Gðt; q; pÞ;

K0ðt − τ; q; zÞKmðτ; z; pÞ dvz

Kmðt; q; pÞ;

ðxÞ〉 dvx ¼ δij

G0ðt − τ; q; zÞKðτ; z; pÞ dvz

(17)

} is a

$$
\sigma\_i(t, p, w) = c\_i(p, w)e^{-\lambda\_i t}
$$

and since σ<sup>i</sup> depend linearly on w, so ciðp;wÞ ¼ ciðpÞw, where ciðpÞ : Ep ! R is a linear function. There exists ~ciðpÞ independent of w such that ciðpÞw ¼ 〈~ciðpÞ;w〉 so that

$$(G(t,q,p)w = \sum\_{i=1}^{\infty} \sigma^{\lambda\_i t} \,\,\psi\_i(q) \langle \tilde{c}\_i(p), w \rangle$$

Consequently, for any β ∈ ΓðEÞ, we have

$$\beta(q) = \lim\_{t \to 0} \int\_M G(t, q, p) \beta(p) \, d\upsilon\_p = \sum\_{k=1}^{\infty} \psi\_k(q) \int\_M \langle \check{c}\_k(p), \beta(p) \rangle \, d\upsilon\_p$$

Moreover, βðqÞ can also be expanded in terms of the ψ<sup>k</sup> basis set,

$$\beta(q) = \sum\_{k=1}^{\infty} \psi\_k(q) \int\_M \langle \psi\_k(p), \beta(p) \rangle \, d\upsilon\_{p^\*}$$

Upon comparing these last two expressions, it is clear that ~ckðpÞ ¼ ψkðpÞ 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


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� dRðMÞ. Define

$$Z^k(M) = \ker\{d : \Lambda^k(M) \to \Lambda^{k+1}(M)\} = \{\alpha \in \Lambda^k(M) \,|\, d\alpha = 0\} \tag{19}$$

$$B^k(M) = \text{ } \text{Im } \{ d : \Lambda^{k-1}(M) \to \Lambda^k(M) \} \!\!/ \!\!/ \!d(\Lambda^{k-1}(M)) \tag{20}$$

Since <sup>d</sup><sup>2</sup> <sup>¼</sup> 0, it follows that <sup>B</sup><sup>k</sup> <sup>ð</sup>M<sup>Þ</sup> <sup>⊂</sup> Zk ðMÞ andthe k-th de Rham cohomology group of M is defined to be

$$H\_{d\mathbb{R}}^k(M) = Z^k(M) / B^k(M) \tag{21}$$

From Eq. (21), construct

$$H\_{dR}^\*(M) = \oplus\_k H\_{dR}^k(M) \tag{22}$$

In 1935, Hodge claimed a theorem, which stated every element in H<sup>k</sup> dRðMÞ 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<sup>k</sup> ðMÞ.

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

$$\mathcal{H}^k(M) = \ker\left\{ d + \delta : \Lambda^k(M) \to \Lambda^\*(M) \right\} = \ker\left\{ \Lambda : \Lambda^k(M) \to \Lambda^k(M) \right\} \tag{23}$$

where Δ ¼ ðd þ δÞ 2 .

Proof: Since <sup>Δ</sup> <sup>¼</sup> <sup>d</sup><sup>δ</sup> <sup>þ</sup> <sup>δ</sup>d, this implies that <sup>Δ</sup>ðΛ<sup>k</sup> <sup>ð</sup>MÞÞ <sup>⊂</sup> <sup>Λ</sup><sup>k</sup> ðMÞ andit is clear that

Hk <sup>ð</sup>M<sup>Þ</sup> <sup>⊂</sup> ker{<sup>d</sup> <sup>þ</sup> <sup>δ</sup> : <sup>Λ</sup><sup>k</sup> ðMÞ ! Λ� <sup>ð</sup>MÞ} <sup>⊂</sup> ker{<sup>Δ</sup> : <sup>Λ</sup><sup>k</sup> ðMÞ ! Λ� <sup>ð</sup>MÞ} <sup>¼</sup> ker {<sup>Δ</sup> : <sup>Λ</sup><sup>k</sup> <sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup> ðMÞ}: To finish the proof, it suffices to show that ker{Δ : Λ<sup>k</sup> <sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup> <sup>ð</sup>MÞ} <sup>⊂</sup> <sup>H</sup><sup>k</sup> ðMÞ. If α∈ker{Δ : Λ<sup>k</sup> <sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup> ðMÞ}, that is Δα ¼ 0, then

$$\begin{aligned} \langle\langle\langle\alpha a,\alpha\rangle\rangle\rangle = \langle\langle(d+\delta)^2\alpha,a\rangle\rangle = \langle\langle(d+\delta)a,(d+\delta)a\rangle\rangle = \langle\langleda,da\rangle\rangle + \langle\langle\delta a,\delta a\rangle\rangle + 2\langle\langleda,\delta a\rangle\rangle \\ = \langle\langle da,da\rangle\rangle + \langle\langle\delta a,\delta a\rangle\rangle = 0 \end{aligned}$$

This implies that <sup>d</sup><sup>α</sup> <sup>¼</sup> <sup>0</sup> and δα <sup>¼</sup> 0, hence <sup>α</sup> <sup>∈</sup> <sup>H</sup><sup>k</sup> ðMÞ.

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


$$
\Lambda^k(M) = H^k(M) + d(\Lambda^{k-1}(M)) + \delta(\Lambda^{k+1}(M))\tag{24}
$$

Proof: By Theorem 2.1, Δ : Λ<sup>k</sup> <sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup> ðMÞ is a Schrödinger operator, so the Hodge theorem applies. Thus H<sup>k</sup> ðMÞ is of finite dimension, so the first holds. The second part of the Hodge theorem is Λ<sup>k</sup> <sup>ð</sup>MÞ ¼ <sup>H</sup><sup>k</sup> <sup>ð</sup>MÞ þ <sup>Δ</sup>ðΛ<sup>k</sup> <sup>ð</sup>MÞÞ. Since <sup>Δ</sup>ðΛ<sup>k</sup> <sup>ð</sup>MÞÞ⊂dðΛ<sup>k</sup>−<sup>1</sup> <sup>ð</sup>MÞÞ þ <sup>δ</sup>ðΛ<sup>k</sup>þ<sup>1</sup> ðMÞÞ, we have Λk <sup>ð</sup>MÞ ¼ <sup>H</sup><sup>k</sup> <sup>ð</sup>MÞ þ <sup>d</sup>ðΛ<sup>k</sup>−<sup>1</sup> <sup>ð</sup>MÞÞ þ <sup>δ</sup>ðΛ<sup>k</sup>þ<sup>1</sup> ðMÞÞ. 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 � : <sup>H</sup><sup>k</sup> <sup>ð</sup>MÞ ! <sup>H</sup><sup>n</sup>−<sup>k</sup> ðMÞ induces an isomorphism

$$H\_{dR}^{k}(M) \simeq H\_{dR}^{n-k}(M) \tag{25}$$

The <sup>k</sup>-th Betti number defined as bkðMÞ ¼ dimH<sup>k</sup> ðM; RÞ also satisfies bkðMÞ ¼ bn<sup>−</sup>kðMÞ for 0 ≤ k ≤ n.
