3. The Minakshisundaran-Pleijel paramatrix

The Hodge theorem has many applications, but one in particular fits here. It is used in

<sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup>

<sup>ð</sup>MÞ=Bk

<sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup>þ<sup>1</sup>

dRðMÞ ¼ Zk

dRðMÞ ¼ <sup>⊕</sup><sup>k</sup> <sup>H</sup><sup>k</sup>

by a unique harmonic form α, one which satisfies both dα ¼ 0 and δα ¼ 0. Denote the set of

<sup>ð</sup>MÞ} <sup>¼</sup> ker {<sup>Δ</sup> : <sup>Λ</sup><sup>k</sup>

<sup>ð</sup>MÞÞ <sup>⊂</sup> <sup>Λ</sup><sup>k</sup>

ðMÞ.

ðMÞ as

<sup>ð</sup>MÞÞ þ <sup>δ</sup>ðΛ<sup>k</sup>þ<sup>1</sup>

ðMÞ ! Λ�

α;α〉〉 ¼ 〈〈ðd þ δÞα;ðd þ δÞα〉〉 ¼ 〈〈dα;dα〉〉 þ 〈〈δα;δα〉〉 þ 2〈〈dα;δα〉〉

ðMÞ ! Λ�

<sup>ð</sup>MÞ} <sup>⊂</sup> ker{<sup>Δ</sup> : <sup>Λ</sup><sup>k</sup>

¼ 〈〈dα;dα〉〉 þ 〈〈δα;δα〉〉 ¼ 0

<sup>ð</sup>MÞ þ <sup>d</sup>ðΛ<sup>k</sup>−<sup>1</sup>

dRðMÞ. Define

<sup>ð</sup>MÞ}≡dðΛ<sup>k</sup>−<sup>1</sup>

ðMÞ andthe k-th de Rham cohomology group of M is

ðMÞj dα ¼ 0} (19)

ðMÞ (21)

dRðMÞ (22)

<sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup>

ðMÞ andit is clear that

<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>

ðMÞÞ (20)

dRðMÞ can be represented

ðMÞ} (23)

<sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup>

<sup>ð</sup>MÞ} <sup>⊂</sup> <sup>H</sup><sup>k</sup>

ðMÞÞ (24)

ðMÞ}:

ðMÞ. If

<sup>ð</sup>MÞ} <sup>≡</sup> {<sup>α</sup> <sup>∈</sup> <sup>Λ</sup><sup>k</sup>

conjunction with the de Rham cohomology group H�

Bk

ðMÞ.

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>

2

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>

2. There is an orthogonal decomposition of Λ<sup>k</sup>

Λk

<sup>ð</sup>MÞ ¼ ker {<sup>d</sup> <sup>þ</sup> <sup>δ</sup> : <sup>Λ</sup><sup>k</sup>

ðMÞ ! Λ�

To finish the proof, it suffices to show that ker{Δ : Λ<sup>k</sup>

ðMÞ}, that is Δα ¼ 0, then

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

<sup>ð</sup>MÞ ¼ <sup>H</sup><sup>k</sup>

ðMÞ is a finite dimensional vector space for k ¼ 0; 1; 2;…; n.

<sup>ð</sup>MÞ ¼ ker{<sup>d</sup> : <sup>Λ</sup><sup>k</sup>

<sup>ð</sup>MÞ ¼ Im {<sup>d</sup> : <sup>Λ</sup><sup>k</sup>−<sup>1</sup>

<sup>ð</sup>M<sup>Þ</sup> <sup>⊂</sup> Zk

Hk

H�

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

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

Zk

Since <sup>d</sup><sup>2</sup> <sup>¼</sup> 0, it follows that <sup>B</sup><sup>k</sup>

74 Manifolds - Current Research Areas

From Eq. (21), construct

harmonic forms as H<sup>k</sup>

where Δ ¼ ðd þ δÞ

α∈ker{Δ : Λ<sup>k</sup>

1. H<sup>k</sup>

<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>

〈〈Δα;α;〉〉 ¼ 〈〈ðd þ δÞ

Hk

Hk

2 .

<sup>ð</sup>MÞ ! <sup>Λ</sup><sup>k</sup>

defined to be

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 <sup>e</sup><sup>−</sup>ρ2=4<sup>t</sup> and parameters <sup>ð</sup>t;p;qÞ∈ð0;∞<sup>Þ</sup> · <sup>M</sup> · <sup>M</sup>, defined by

$$H\_{\circ}(t, q, p) = \frac{1}{\left(4\pi t\right)^{n/2}} e^{-\rho^2/4t} \sum\_{k=0}^{\infty} t^k \, u\_k(p, q) : E\_p \to E\_q \tag{26}$$

In Eq. (26), the function <sup>ρ</sup> <sup>¼</sup> <sup>ρ</sup>ðp;q<sup>Þ</sup> is the metric distance between <sup>p</sup> and <sup>q</sup> in <sup>M</sup>, Ep <sup>¼</sup> <sup>π</sup><sup>−</sup><sup>1</sup>ðp<sup>Þ</sup> is the fiber of E over p and ukðp;qÞ : Ep ! Eq are R-linear map.

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

$$H\left(\frac{\partial}{\partial t} + \Delta\_q\right)H\_{\circ}(t, q, p)w = 0\tag{27}$$

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 ðx1;…;xnÞ, then the function ρðp;qÞ is

$$\rho(p,q) = \sqrt{\mathbf{x}\_1^2 + \dots + \mathbf{x}\_n^2} \tag{28}$$

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

$$\mathcal{g}\_{ij} = \left\langle \frac{\partial}{\partial \alpha\_i}, \frac{\partial}{\partial \alpha\_j} \right\rangle, \qquad \mathcal{G} = \det(\mathcal{g}\_{ij}) \tag{29}$$

and define the differential operator

$$
\hat{\mathfrak{g}} = \sum\_{k=1}^{n} \mathfrak{x}\_k \stackrel{\partial}{\to} \hat{\mathfrak{g}}\_k
$$

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

$$\begin{split} \frac{\partial}{\partial t} H\_{\circ \circ} (t, p, q) \text{w} &= \frac{1}{\left(4\pi t\right)^{n/2}} e^{-\rho^{2}/4t} \left\{ \left(\frac{\rho^{2}}{4t^{2}} - \frac{n}{2t}\right) \sum\_{k=0}^{\circ} t^{k} \, u\_{k}(p, q) \text{w} + \sum\_{k=0}^{\circ} kt^{k-1} u\_{k}(p, q) \text{w} \right\} \\ &= \frac{1}{\left(4\pi t\right)^{n/2}} e^{-\rho^{2}/4t} \sum\_{k=0}^{\circ} \left\{ \frac{\rho^{2}}{4t^{2}} - \frac{n}{2t} + \frac{k}{t} \right\} t^{k} u\_{k}(p, q) \text{w} \end{split} \tag{30}$$

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

$$\Phi(\rho) = \frac{e^{-\rho^2/4t}}{(4\pi t)^{n/2}}\tag{31}$$

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

$$e\_i(p) = \frac{9}{9x\_i}|\_p$$

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

$$\begin{aligned} \Delta\_0 H\_{\curvearrowright}(t, p, q)\varpi &= (\Delta\_0 \bullet \mathfrak{D}) \cdot \left(\sum\_{k=0}^{\infty} t^k u\_k(p, q)\varpi \right) \\ + 2\sum\_{a=1}^n (\varepsilon\_a \bullet \mathfrak{D}) \cdot \nabla\_{\varepsilon\_a} \left(\sum\_{k=0}^{\infty} t^k u\_k(p, q)\varpi \right) &+ \mathfrak{D} \cdot \Delta\_0 \left(\sum\_{k=0}^{\infty} t^k u\_k(p, q)\varpi \right) \end{aligned} \tag{32}$$

The individual components of (32) can be calculated as follows; since Φ is a function ∇ea Φ ¼ eaΦ and so

$$\begin{aligned} \varepsilon\_a \Phi(\rho) &= \operatorname{\boldsymbol{\phi}}^\circ(\rho) e\_a(\rho), \\ \Delta\_0 \Phi = \sum\_a \{e\_a \, \varepsilon\_a \Phi(\rho) - (\nabla\_{e\_a} e\_a) \Phi(\rho)\} &= \operatorname{\boldsymbol{\phi}}^\*(\rho) \cdot \sum\_a (e\_a \rho)^2 + \operatorname{\boldsymbol{\phi}}^\circ(\rho) \cdot \Delta\_0 \rho, \\ \operatorname{\boldsymbol{\phi}}^\circ(\rho) &= -\frac{\rho}{2t} \Phi(\rho), \\ \operatorname{\boldsymbol{\phi}}^\circ(\rho) &= \left(\frac{\rho^2}{4t^2} - \frac{1}{2t}\right) \Phi(\rho) \end{aligned} \tag{33}$$

Consequently,

#### Spectral Theory of Operators on Manifolds http://dx.doi.org/10.5772/64611 77

$$e\_a \rho = \frac{\varkappa\_a}{\rho}, \qquad \sum\_a (e\_a \rho)^2 = 1, \qquad \Lambda\_0 \rho = \frac{n-1}{\rho} + \frac{1}{\rho} \stackrel{\frown}{\partial} \quad \log \sqrt{G}$$

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

gij <sup>¼</sup> <sup>∂</sup> ∂xi ; ∂ ∂xj 

> ∂ ^ ¼ ∑ n k¼1 xk ∂ ∂xk

<sup>−</sup>ρ2=4<sup>t</sup> ρ<sup>2</sup>

<sup>−</sup>ρ2=4<sup>t</sup> ∑ ∞ k¼0

4t 2 − n 2t 

The notion of the heat operator (15) on Eq. (26) is worked out one term at a time. First, the

ρ2 4t 2 − n 2t þ k t

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

<sup>Φ</sup>ðρÞ ¼ <sup>e</sup><sup>−</sup>ρ2=4<sup>t</sup> ð4πtÞ

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

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

Δ0H∞ðt;p;qÞw ¼ ðΔ<sup>0</sup> ΦÞ �

 ∑ ∞ k¼0 t k ukðp;qÞw

eiðpÞ ¼ <sup>∂</sup> ∂xi j p

> ∑ ∞ k¼0 t

þ Φ � Δ<sup>0</sup>

ðρÞeaðρÞ;

ðρÞ � ∑ a ðeaρÞ

ΦðρÞ

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

eaΦðρÞ ¼ <sup>Φ</sup>′

<sup>ð</sup>ρÞ ¼ <sup>ρ</sup><sup>2</sup> 4t 2 − 1 2t 

<sup>ð</sup>ρÞ ¼ <sup>−</sup> <sup>ρ</sup> 2t ΦðρÞ;

{ea eaΦðρÞ−ð∇ea eaÞΦðρÞ} <sup>¼</sup> <sup>Φ</sup>″

Φ′

Φ″

kukðp;qÞ<sup>w</sup>

 ∑ ∞ k¼0 t <sup>2</sup> <sup>þ</sup> <sup>Φ</sup>′

ðρÞ � Δ0ρ;

kukðp;qÞ<sup>w</sup>

(32)

(33)

∑ ∞ k¼0 t

<sup>k</sup> ukðp; <sup>q</sup>Þ<sup>w</sup> <sup>þ</sup> <sup>∑</sup>

t k ukðp;qÞw

∞ k¼0 ktk<sup>−</sup><sup>1</sup>

<sup>n</sup>=<sup>2</sup> (31)

ukðp;qÞw

(30)

and define the differential operator

76 Manifolds - Current Research Areas

derivative with respect to t is calculated

<sup>H</sup>∞ðt;p;qÞ<sup>w</sup> <sup>¼</sup> <sup>1</sup>

þ2 ∑ n a¼1

Δ0Φ ¼ ∑ a

∇ea Φ ¼ eaΦ and so

Consequently,

ðeaΦÞ � ∇ea

ð4πtÞ

<sup>¼</sup> <sup>1</sup> ð4πtÞ

<sup>n</sup>=<sup>2</sup> e

<sup>n</sup>=<sup>2</sup> e

∂ ∂t

follows:

; G ¼ detðgijÞ (29)

$$
\Delta\_0 \: \Phi = \mathcal{O}(\rho) \left( \left( \frac{\rho^2}{4t^2} - \frac{1}{2t} \right) - \frac{1}{2t} (n - 1 - \hat{\partial} \log \sqrt{G}) \right) \tag{34}
$$

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,

$$\begin{split} 2\sum\_{a=1}^{n} \left( \mathcal{e}\_{\mathtt{d}} \, \mathcal{O} \right) \cdot \nabla\_{\mathcal{e}\_{x}} \left( \sum\_{k=0}^{\infty} \, \mathsf{t}^{k} u\_{k}(p,q) w \right) &= 2 \, \mathsf{O}'(\rho) \sum\_{a=1}^{n} \, \frac{\mathsf{X}\_{\mathtt{d}}}{\rho} \cdot \nabla\_{\mathcal{e}\_{x}} \left( \sum\_{k=0}^{\infty} \, \mathsf{t}^{k} u\_{k}(p,q) w \right) \\ &= -\frac{\rho}{t} \, \mathsf{O}'(\rho) \, \nabla\_{\hat{\mathsf{S}}\_{\hat{\mathsf{I}}}/\rho} \left( \sum\_{k=0}^{\infty} \, \mathsf{t}^{k} u\_{k}(p,q) w \right) \end{split} \tag{35}$$

Substituting these results into (32), it follows that

$$\Delta\_0 \, H\_{\approx}(t, q, p) = \mathcal{O}(\rho) \left[ \frac{\rho^2}{4t^2} - \frac{1}{2t} - \frac{1}{2t} (n - 1 - \overset{\frown}{\partial} \log \sqrt{G}) - \frac{\rho}{t} \nabla\_{\mathbb{S}^\times/\rho} + \Delta\_0 \right] \sum\_{m=0}^\infty t^m u\_m(p, q) w \tag{36}$$

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

$$\begin{aligned} \left(\frac{\partial}{\partial t} - \Delta\_0 - F\right) H\_m(t, q, p) w &= \mathcal{O}\left[\left(\nabla\_\delta + \frac{1}{4G} \overset{\wedge}{\partial} G\right) \cdot \frac{1}{t} u\_0(p, q) w + \sum\_{k=1}^m \left[\left(\nabla\_\partial^\wedge + k + \frac{1}{4G} \overset{\wedge}{\partial} G\right) u\_k(p, q) w + \sum\_{k=1}^m \left(\nabla\_\partial^\wedge + k + \frac{1}{4G} \overset{\wedge}{\partial} G\right) w\right] \right] \end{aligned} \tag{37}$$
 
$$- (\Delta\_0 + F) u\_{k-1}(p, q) w \bigg| \int\_{\delta}^{\delta - 1} \tag{38}$$

This is summarized in the following Lemma.

Lemma 3.1. Heat equation (27) for H∞ðt;p;qÞ is equivalent to

$$\left(\nabla\_{\hat{\partial}} + k + \frac{1}{4G}\hat{\partial}\,G\right)u\_k(p,q)w = (\Delta\_0 + F)\,\mu\_{k-1}(p,q)w\tag{38}$$

for all k ¼ 0; 1; 2;… and Eq. (38) is initialized with u<sup>−</sup>1ðp;qÞ ¼ 0.

In fact, for fixed p ∈ M and w ∈ Ep, 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 uðp;qÞw, which depends linearly on w. Then, umðp;qÞ : Ep ! Eq and the Minakshisundaram-Pleijel parametrix for heat operator (Eq. 15) is defined by

$$H\_{\curvearrowright}(t, p, q) = \frac{1}{(4\pi t)^{n/2}} e^{-\rho^2/4t} \sum\_{m=0}^{\curvearrowright} t^m u\_m(p, q) : E\_p \longrightarrow E\_q \tag{39}$$

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

$$H\_N(t, q, p) = \frac{1}{\left(4\pi t\right)^{n/2}} e^{-\rho^2/4t} \sum\_{m=0}^N t^m u\_m(p, q) : E\_p \longrightarrow E\_q \tag{40}$$

Theorem 3.1. Choose a smooth function φ : M · M ! M and let G0ðt;q;pÞ ¼ φðq;pÞHNðt;q;pÞ. Then <sup>G</sup>0ðt;q;p<sup>Þ</sup> is a <sup>k</sup>-th initial solution of the heat operator (15), where <sup>k</sup> <sup>¼</sup> ⌊ <sup>N</sup> <sup>2</sup> − <sup>n</sup> <sup>4</sup> ⌋ and⌊z⌋ is the greatest integer less than or equal to z.

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

$$\left(\frac{\partial}{\partial t} - \Delta\_0 - F\right) H\_N(t, q, p) w = -\frac{1}{\left(4\pi t\right)^{n/2}} e^{-\rho^2/4t} t^{N-\frac{\pi}{2}} (\Delta\_0 + F) \iota\_N(p, q) w \tag{41}$$

and uNðp;q<sup>Þ</sup> is <sup>C</sup><sup>∞</sup> with respect to <sup>p</sup> and <sup>q</sup>. Since <sup>t</sup> N−<sup>n</sup> <sup>2</sup> e<sup>−</sup>ρ2=4<sup>t</sup> is C<sup>k</sup> ð½0;∞Þ · M · MÞ, hence <sup>H</sup>ðϕðp;qÞHNðt;q;pÞÞ∈C<sup>k</sup> ð½0;∞Þ · M · MÞ. Consider integrating G<sup>0</sup> against ψðs;βÞ,

$$\int\_{M} G\_{0}(t, q, \mathbf{s}) \psi(\mathbf{s}, \boldsymbol{\beta}) \, d\upsilon\_{s} = \sum\_{m=0}^{N} t^{m} \int\_{M} \frac{1}{(4\pi t)^{n/2}} e^{-\rho^{2}/4t} \psi(q, \mathbf{s}) \mu\_{m}(\mathbf{s}, q) \psi(\mathbf{s}, \boldsymbol{\beta}) \, d\upsilon\_{s} \tag{42}$$

The integral of Eq. (42) over <sup>M</sup> can be broken up into an integral over Qqð<sup>E</sup> <sup>2</sup>Þ¼fs ∈ Mjρðq;sÞ <sup>&</sup>lt; <sup>E</sup>=2<sup>g</sup> anda second integral over the set <sup>M</sup>−Mqð<sup>E</sup> <sup>2</sup>Þ. On the latter set, the limit converges uniformly hence

$$\lim\_{t \to \infty} \frac{e^{-\rho^2/4t}}{(4\pi t)^{n/2}} = 0$$

To estimate the remaining integral, choose a normal coordinate system at q and denote the integration coordinates as ðs1;…;snÞ, then the integrand of Eq. (42) is given as

$$\frac{1}{\left(4\pi t\right)^{n/2}} e^{-|s|^2/4t} \,\,\phi(q,s) u\_m(s,q) \psi(s,\beta) \sqrt{\det(\frac{\partial}{\partial s\_i}, \frac{\partial}{\partial s\_j})} \,\,\, ds\_1 \cdots ds\_n$$

Therefore, in the limit using Definition 2.4,

$$\lim\_{t \to 0} \int\_{M(\epsilon/2)} \frac{1}{\left(4\pi t\right)^{n/2}} e^{-\rho^2/4t} \,\phi(q, \mathbf{s}) \mu\_m(\mathbf{s}, q) \psi(\mathbf{s}, \boldsymbol{\beta}) \,d\upsilon\_s = \mu\_m(q, q)\psi(q, \boldsymbol{\beta})$$

This result implies that

#### Spectral Theory of Operators on Manifolds http://dx.doi.org/10.5772/64611 79

$$\lim\_{t \to 0} \int\_M G\_0(t, q, \mathbf{s}) \psi(\mathbf{s}, \boldsymbol{\beta}) \, d\mathbf{v}\_s = \sum\_{m=0}^N \lim\_{t \to 0} t^m u\_m(q, q) \psi(q, \boldsymbol{\beta}) = \psi(q, \boldsymbol{\beta}) u\_0(q, q) = \psi(q, \boldsymbol{\beta}) \tag{43}$$

The convergence here is uniform.

<sup>H</sup>∞ðt;p;qÞ ¼ <sup>1</sup>

HNðt;q;pÞ ¼ <sup>1</sup>

greatest integer less than or equal to z.

∂ ∂t −Δ0−F � �

<sup>H</sup>ðϕðp;qÞHNðt;q;pÞÞ∈C<sup>k</sup>

78 Manifolds - Current Research Areas

uniformly hence

ð M

From the previous calculation, it holds that

and uNðp;q<sup>Þ</sup> is <sup>C</sup><sup>∞</sup> with respect to <sup>p</sup> and <sup>q</sup>. Since <sup>t</sup>

G0ðt;q;sÞψðs;βÞ dvs ¼ ∑

<sup>&</sup>lt; <sup>E</sup>=2<sup>g</sup> anda second integral over the set <sup>M</sup>−Mqð<sup>E</sup>

1 ð4πtÞ

Therefore, in the limit using Definition 2.4,

MðE=2Þ

ð

lim t!0

This result implies that

<sup>n</sup>=<sup>2</sup> e −jsj 2

> 1 ð4πtÞ

<sup>n</sup>=<sup>2</sup> e

ð4πtÞ

ð4πtÞ

Then <sup>G</sup>0ðt;q;p<sup>Þ</sup> is a <sup>k</sup>-th initial solution of the heat operator (15), where <sup>k</sup> <sup>¼</sup> ⌊ <sup>N</sup>

HNðt;q;pÞ<sup>w</sup> <sup>¼</sup> <sup>−</sup> <sup>1</sup>

N m¼0 t m ð M

The integral of Eq. (42) over <sup>M</sup> can be broken up into an integral over Qqð<sup>E</sup>

limt!∞

integration coordinates as ðs1;…;snÞ, then the integrand of Eq. (42) is given as

<sup>=</sup>4<sup>t</sup> <sup>ϕ</sup>ðq;sÞumðs;qÞψðs;β<sup>Þ</sup>

<sup>n</sup>=<sup>2</sup> e

<sup>n</sup>=<sup>2</sup> e

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

<sup>−</sup>ρ2=4<sup>t</sup> ∑ ∞ m¼0 t

<sup>−</sup>ρ2=4<sup>t</sup> ∑ N m¼0 t

Theorem 3.1. Choose a smooth function φ : M · M ! M and let G0ðt;q;pÞ ¼ φðq;pÞHNðt;q;pÞ.

Proof: Clearly, G<sup>0</sup> is a linear map of vector spaces andis continuous and C<sup>∞</sup> in all parameters.

ð4πtÞ

<sup>n</sup>=<sup>2</sup> e −ρ2=4t t N − <sup>n</sup>

ð½0;∞Þ · M · MÞ. Consider integrating G<sup>0</sup> against ψðs;βÞ,

1 ð4πtÞ

e<sup>−</sup>ρ2=4<sup>t</sup> ð4πtÞ

To estimate the remaining integral, choose a normal coordinate system at q and denote the

<sup>n</sup>=<sup>2</sup> ¼ 0

s

<sup>−</sup>ρ2=4<sup>t</sup> <sup>ϕ</sup>ðq;sÞumðs;qÞψðs;β<sup>Þ</sup> dvs <sup>¼</sup> umðq;qÞψðq;β<sup>Þ</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det〈 <sup>∂</sup> ∂si ; ∂ ∂sj 〉

<sup>n</sup>=<sup>2</sup> e −ρ2=4t

N−<sup>n</sup>

<sup>2</sup> e<sup>−</sup>ρ2=4<sup>t</sup> is C<sup>k</sup>

mumðp;q<sup>Þ</sup> : Ep ! Eq (39)

mumðp;q<sup>Þ</sup> : Ep ! Eq (40)

<sup>2</sup> − <sup>n</sup>

<sup>2</sup> ðΔ<sup>0</sup> þ FÞuNðp;qÞw (41)

ψðq;sÞumðs;qÞψðs;βÞ dvs (42)

<sup>2</sup>Þ. On the latter set, the limit converges

ds1⋯dsn

ð½0;∞Þ · M · MÞ, hence

<sup>2</sup>Þ¼fs ∈ Mjρðq;sÞ

<sup>4</sup> ⌋ and⌊z⌋ is the

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 andE a vector bundle over M with inner product and metric Riemannian connection. Let Gðt;q;pÞ be the heat kernel or fundamental solution for heat operator (Eq. (15)) and (Eq. (39)) the MP parametrix. Then as t ! 0, Gðt;p;pÞ has the asymptotic expansion Gðt;p;pÞ eH∞ðt;p;pÞ,

that is, for any N > 0, it is the case that

$$G(t, p, p) - \frac{1}{\left(4\pi t\right)^{n/2}} \sum\_{m=0}^{N} t^m u\_m(p, p) = O\left(t^{N\frac{n}{2}}\right) \tag{44}$$

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

$$\lim\_{t \to 0} \frac{\xi}{t^{N-\frac{\pi}{2}}} = 0$$

Proof: It suffices to prove the theorem for any large N. Let G0ðt;q;pÞ ¼ ϕðq;pÞHNðt;q;pÞ as in Theorem 3.2. The conclusion of the theorem is equivalent to the statement

$$G(t, p, p) - G\_0(t, p, p) = O\left(t^{N \frac{n}{2}}\right)$$

From the previous theorem and existence and regularity of the fundamental solution, the result G of Levi iteration initialized by G<sup>0</sup> is exactly the fundamental solution. Equality (Eq. (41)) means that there exists a constant A such that for any t∈ð0;TÞ,

$$|K\_0(t, q, p)| = |\left(\frac{\partial}{\partial t} + \Delta\right) G\_0(t, q, p)| \le A t^{N - \frac{n}{2}}$$

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

$$|K\_1(t, q, p)| \le \int\_0^t d\tau \int\_M |K\_0(t - \tau, q, s) K\_0(\tau, s, p)| \, \, d\upsilon\_s$$

$$\le \int\_0^t [A^2(t - \tau)^{N - \frac{n}{2}} \tau^{N - \frac{n}{2}} \upsilon(M)] \, d\tau \le \int\_0^t A^2 T^{N - \frac{n}{2}} \tau^{N - \frac{n}{2}} \upsilon(M) \, d\tau \le AB \frac{t^{N - \frac{n}{2} + 1}}{N - \frac{n}{2} + 1}$$

We have set <sup>B</sup> <sup>¼</sup> <sup>A</sup> � <sup>T</sup><sup>N</sup>−<sup>n</sup> <sup>2</sup> vðMÞ. Exactly the same procedure applies to jK2ðt;q;pÞj. Based on the pattern established this way, induction implies that the following bound results

$$|K\_m(t,q,p)| \leq A \quad \cdot B^m \frac{t^{N - \frac{n}{2} + m}}{\left(N - \frac{n}{2} + 1\right)\left(N - \frac{n}{2} + 2\right)\cdots\left(N - \frac{n}{2} + m\right)} \leq A \cdot B^m \frac{t^m}{m!} \; t^{N - \frac{n}{2}}$$

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

$$|\check{K}(t,q,p)| \leq \sum\_{m=0}^{\infty} |K\_m(t,q,p)| \leq A \cdot e^{\operatorname{Bt}} t^{N-\frac{\omega}{2}}$$

Using this bound, the required estimate is obtained,

$$|G(t,q,p) - G\_0(t,q,p)| \leq \left| \int\_0^t d\tau \int\_M d\upsilon\_z G\_0(t-\tau,q,z) \check{K} \ (\tau,z,p) \right|$$

$$\leq \int\_0^t d\tau \int\_M \frac{e^{-p^2/4(t-\tau)}}{\left(4\pi(t-\tau)\right)^{n/2}} A \cdot e^{B\tau} \cdot \tau^{N-\frac{\pi}{2}} d\upsilon\_s$$

$$\leq M\_n A e^{Bt} \int\_0^t \tau^{N-\frac{\pi}{2}} d\tau \ v(M) = \frac{1}{N-\frac{n}{2}+1} M\_n A \cdot e^{Bt} v(M) t^{N-\frac{n}{2}+1}$$

This finishes the proof.

Now if all the Hodge theorem is used, formal expressions for the index can be obtained. Suppose D : ΓðEÞ ! ΓðFÞ is an operator such that D� D and DD� are Schrödinger operators andD� is the adjoint of D. Suppose the operators D� D : ΓðEÞ ! ΓðEÞ and DD� : ΓðFÞ ! ΓðEÞ are defined, so they are self-adjoint and have nonnegative real eigenvalues. Then the spaces ΓμðEÞ and ΓμðFÞ can be defined this way

$$\Gamma\_{\mu}(E) = \{ \varphi \in \Gamma(E) | D^\* D \varphi = \mu \varphi \}, \qquad \Gamma\_{\mu}(F) = \{ \varphi \in \Gamma(F) | D D^\* \varphi = \mu \varphi \} \tag{45}$$

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

$$
\Gamma\_0(E) = \ker \{ D : \Gamma(E) \to \Gamma(F) \}, \qquad \Gamma\_0(F) = \ker \{ D^\* : \Gamma(F) \to \Gamma(E) \}
$$

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

$$\text{Ind } D = \text{dim } \ker D\\\text{-dim } \ker D^\* = \dim \Gamma\_0(E)\\\text{-dim } \Gamma\_0(F)$$

Definition 3.2. For the Schrödinger operator <sup>Δ</sup>, let <sup>e</sup><sup>−</sup>t<sup>Δ</sup> : <sup>Γ</sup>ðEÞ ! <sup>Γ</sup>ðEÞ, for <sup>t</sup> <sup>&</sup>gt; 0 be defined as

$$(e^{-t\Lambda}\varphi)(q) = \int\_M G(t, q, p)\varphi(p)dv\_p\tag{46}$$

where Gðt;q;pÞ is the fundamental solution of heat operator (Eq. (15)).

Let 0 ≤ λ<sup>1</sup> ≤ λ<sup>2</sup> ≤ ⋯ ! ∞ be the eigenvalues of the operator Δ and {ψ1;ψ2;…} the corresponding eigenfunctions. Intuitively, the trace of e<sup>−</sup>t<sup>Δ</sup> is defined as

#### Spectral Theory of Operators on Manifolds http://dx.doi.org/10.5772/64611 81

$$\text{tr } \boldsymbol{e}^{-t\boldsymbol{\Lambda}} = \sum\_{k=1}^{\infty} \left\langle \left\langle \boldsymbol{e}^{-t\boldsymbol{\Lambda}} \,\,\psi\_{k}, \psi\_{k} \right\rangle \right\rangle \tag{47}$$

This is clearly <sup>∑</sup>ke<sup>−</sup>λ<sup>k</sup> <sup>t</sup> or <sup>∑</sup><sup>μ</sup> <sup>e</sup><sup>−</sup>t<sup>μ</sup> dim ΓμðEÞ, so the definition of tr is well-defined if and only if

$$\sum\_{k} e^{-\lambda\_k t} < \infty \tag{48}$$

Theorem 3.3. For any p;q∈M, let {e1ðpÞ;…;eNðpÞ} and {f <sup>1</sup>ðqÞ;…;f <sup>N</sup>ðqÞ} be orthonormal bases on Ep and Eq, respectively, then the following two results hold for t > 0,

$$\begin{aligned} (a) \qquad & \int\_{M} \int\_{a} \sum\_{b=1}^{N} \left< G(t, q, p) e\_{a}(p) f\_{b}(q) \right>^{2} d\upsilon\_{q} d\upsilon\_{p} < \Leftrightarrow, \\\ (b) \qquad & \sum\_{k=1}^{\infty} e^{2\lambda\_{k}t} < \int\_{M} \int\_{a, b=1}^{N} \left< G(t, q, p) e\_{a}(p) f\_{b}(q) \right>^{2} d\upsilon\_{q} d\upsilon\_{p} < \Leftrightarrow \end{aligned} \tag{49}$$

Proof: When t > 0, Gðt;q;pÞ is continuous and hence satisfies (a). For and w ∈ ΓðEÞ, Theorem 2.5 yields the following expansion for Gðt;q;pÞ ∈ ΓðEÞ, hence the Parseval equality yields

$$\int\_{M} \left| G(t, q, p)w \right|^2 d\upsilon\_{\emptyset} = \sum\_{k=1}^{\infty} e^{-2\lambda\_{k}t} \langle \psi\_{k}(p), w \rangle^2$$

Replacing w by the basis element eaðpÞ, this implies that

$$\sum\_{a=1}^{N} \int\_{M} |G(t, q, p) e\_a(p)|^2 \, d\upsilon\_q$$

$$\xi = \sum\_{a=1}^{N} \sum\_{k=1}^{\infty} e^{-2\lambda\_k t} \langle \psi\_k(p), e\_a(p) \rangle^2 = \sum\_{k=1}^{\infty} \sum\_{a=1}^{N} e^{-2\lambda\_k t} \langle \psi\_k(p), e\_a(p) \rangle^2 = \sum\_{k=1}^{\infty} e^{-2\lambda\_k t} \langle \psi\_k(p), \psi\_k(p) \rangle^2$$

Then for any m, it follows that

<sup>j</sup>Kmðt;q;pÞj <sup>≤</sup> <sup>A</sup> � Bm <sup>t</sup>

Using this bound, the required estimate is obtained,

≤ ðt 0 dτ ð M

ðt 0 τ<sup>N</sup>−<sup>n</sup>

Suppose D : ΓðEÞ ! ΓðFÞ is an operator such that D�

andD� is the adjoint of D. Suppose the operators D�

ΓμðEÞ ¼ {ϕ ∈ ΓðEÞjD�

≤ MnAeBt

ΓμðEÞ and ΓμðFÞ can be defined this way

This finishes the proof.

80 Manifolds - Current Research Areas

jGðt;q;pÞ−G0ðt;q;pÞj ≤ j

N− <sup>n</sup> <sup>2</sup> <sup>þ</sup> <sup>1</sup> � � <sup>N</sup><sup>−</sup> <sup>n</sup>

<sup>j</sup>K<sup>~</sup> <sup>ð</sup>t;q;pÞj <sup>≤</sup> <sup>∑</sup>

N−<sup>n</sup> <sup>2</sup>þm

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

∞ m¼0

> ðt 0 dτ ð M

e<sup>−</sup>ρ2=4ðt−τ<sup>Þ</sup> ð4πðt−τÞÞ<sup>n</sup>=<sup>2</sup> <sup>A</sup> � <sup>e</sup>

<sup>2</sup> <sup>d</sup><sup>τ</sup> <sup>v</sup>ðMÞ ¼ <sup>1</sup>

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

ðe

eigenfunctions. Intuitively, the trace of e<sup>−</sup>t<sup>Δ</sup> is defined as

where Gðt;q;pÞ is the fundamental solution of heat operator (Eq. (15)).

<sup>−</sup><sup>t</sup>ΔϕÞðqÞ ¼

N− n <sup>2</sup> <sup>þ</sup> <sup>1</sup>

Now if all the Hodge theorem is used, formal expressions for the index can be obtained.

are defined, so they are self-adjoint and have nonnegative real eigenvalues. Then the spaces

Γ0ðEÞ ¼ ker{D : ΓðEÞ ! ΓðFÞ}; Γ0ðFÞ ¼ ker{D� : ΓðFÞ ! ΓðEÞ}

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

Definition 3.2. For the Schrödinger operator <sup>Δ</sup>, let <sup>e</sup><sup>−</sup>t<sup>Δ</sup> : <sup>Γ</sup>ðEÞ ! <sup>Γ</sup>ðEÞ, for <sup>t</sup> <sup>&</sup>gt; 0 be defined as

Let 0 ≤ λ<sup>1</sup> ≤ λ<sup>2</sup> ≤ ⋯ ! ∞ be the eigenvalues of the operator Δ and {ψ1;ψ2;…} the corresponding

ð M

Ind D ¼ dim ker D−dim ker D� ¼ dim Γ0ðEÞ−dim Γ0ðFÞ

Dϕ ¼ μϕ}; ΓμðFÞ ¼ {ϕ∈ΓðFÞjDD�

<sup>2</sup> <sup>þ</sup> <sup>2</sup> � �<sup>⋯</sup> <sup>N</sup><sup>−</sup> <sup>n</sup>

jKmðt;q;pÞj ≤ A � e

<sup>2</sup> <sup>þ</sup> <sup>m</sup> � � <sup>≤</sup> <sup>A</sup> � <sup>B</sup><sup>m</sup> <sup>t</sup>

Btt N−<sup>n</sup> 2

dvzG0ðt−τ;q;zÞK<sup>~</sup> <sup>ð</sup>τ;z;pÞj

<sup>2</sup> dvs

BtvðMÞ<sup>t</sup>

N−<sup>n</sup> <sup>2</sup>þ1

D and DD� are Schrödinger operators

D : ΓðEÞ ! ΓðEÞ and DD� : ΓðFÞ ! ΓðEÞ

Gðt;q;pÞϕðpÞdvp (46)

ϕ ¼ μϕ} (45)

<sup>B</sup><sup>τ</sup> � <sup>τ</sup><sup>N</sup>−<sup>n</sup>

MnA � e

m <sup>m</sup>! <sup>t</sup> N−<sup>n</sup> 2

$$\begin{split} \sum\_{k=1}^{m} e^{-2\lambda\_{k}t} &= \sum\_{k=1}^{m} \int\_{M} e^{-2\lambda\_{k}t} \langle \psi\_{k}(p), \psi\_{k}(p) \rangle \, d\upsilon\_{p} \le \int\_{M} \sum\_{k=1}^{\infty} e^{-2\lambda\_{k}t} \langle \psi\_{k}(p), \psi\_{k}(p) \rangle \, d\upsilon\_{p} \\ = \int\_{M} d\upsilon\_{p} \int\_{M} \sum\_{a=1}^{N} \left| G(t, q, p) \varepsilon\_{a}(p) \right|^{2} \, d\upsilon\_{q} = \int\_{M} \int\_{M} \sum\_{a,b=1}^{N} \left\langle G(t, q, p) \varepsilon\_{a}(p), f\_{b}(q) \right\rangle^{2} \, d\upsilon\_{q} d\upsilon\_{p} < \infty. \end{split}$$

Theorem 3.4. For any t > 0,

$$\text{tr } \left( e^{-t\Lambda} \right) = \int\_M tr \, G(t, p, p) \, dv\_p \tag{50}$$

Proof: From Theorem 2.2, it follows that

$$\begin{aligned} \text{tr}\,G(t,p,p) &= \sum\_{a=1}^{N} \left< G(t,p,p)e\_a(p), e\_a(p) \right> = \sum\_{a=1}^{N} \left< \sum\_{k=1}^{\infty} e^{-t\lambda\_k} \langle \psi\_k(p)e\_a(p) \rangle \psi\_k(p), e\_a(p) \right> \\ &= \sum\_{a=1}^{N} \sum\_{k=1}^{\infty} e^{-t\lambda\_k} \langle \psi\_k(p), e\_a(p) \rangle^2 = \sum\_{k=1}^{\infty} e^{-t\lambda\_k} \langle \psi\_k(p), \psi\_k(p) \rangle^2 \end{aligned}$$

Integrating this on both sides, it is found that

$$\int\_{M} \text{tr } G(t, p, p) \, d\upsilon\_{p} = \int\_{M} \sum\_{k=1}^{\infty} e^{-t\lambda\_{k}} \langle \psi\_{k}(p), \psi\_{k}(p) \rangle^{2} \, d\upsilon\_{p} = \sum\_{k=1}^{\infty} e^{-t\lambda\_{k}} = \text{tr } (e^{-t\Delta})^{2}$$

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

$$\lim\_{t \to \infty} \quad \text{tr} \ e^{-t\Lambda} = \sum\_{k=1}^{\infty} \lim\_{t \to \infty} e^{-t\lambda\_k} = \text{dim } \Gamma\_0(E) \tag{51}$$

In fact, as t ! 0, the equality

$$G(t, p, p) = \frac{1}{(4\pi t)^{n/2}} + O\left(\frac{1}{t^{n/2}}\right).$$

and the previous theorem imply that lim<sup>t</sup>!<sup>0</sup> tr <sup>e</sup><sup>−</sup>t<sup>Δ</sup> <sup>¼</sup> <sup>∞</sup>.
