4. An application of the expansions: the Gauss Bonnet theorem

As far as Ind ðDÞ is concerned, it is the case for all t > 0 that,

$$\text{Ind}\ (D) = \text{tr }e^{-tD'D} - \text{tr }e^{-tDD'} = \int\_M \text{tr }\mathcal{G}\_+(t,p,p) \, d\upsilon\_p - \int\_M \text{tr }\mathcal{G}\_-(t,p,p) \, d\upsilon\_p$$

by Theorem 3.5, where G�ðt;p;pÞ are the fundamental solutions of ∂<sup>t</sup> þ D� D and ∂<sup>t</sup> þ DD� . As t ! 0, Theorem 3.2 assumes the form

$$G\_{\pm}(t,p,p) \therefore H\_{\approx}^{\pm}(t,p,p) = \frac{1}{\left(4\pi t\right)^{n/2}} \sum\_{m=0}^{\infty} t^{m} u\_{\pm m}(p,p)$$

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

$$\sum\_{k} e^{-\lambda\_k t} = \frac{1}{\left(4\pi t\right)^{n/2}} \sum\_{k=0}^{\infty} \int\_{M} \mu\_k(\mathbf{x}, \mathbf{x}) \, d\upsilon\_{\mathbf{x}} \tag{52}$$

Proof:

tr Gðt;p;pÞ ¼ ∑

ð M

82 Manifolds - Current Research Areas

In fact, as t ! 0, the equality

Ind ðDÞ ¼ tr e

t ! 0, Theorem 3.2 assumes the form

fore,

N a¼1

¼ ∑ N a¼1 ∑ ∞ k¼1

Integrating this on both sides, it is found that

tr Gðt;p;pÞ dvp ¼

〈Gðt;p;pÞeaðpÞ;eaðpÞ〉 ¼ ∑

<sup>e</sup><sup>−</sup>tλ<sup>k</sup> 〈ψkðpÞ;eaðpÞ〉

<sup>−</sup>t<sup>Δ</sup> <sup>¼</sup> <sup>∑</sup> ∞ k¼1

<sup>G</sup>ðt;p;pÞ ¼ <sup>1</sup>

4. An application of the expansions: the Gauss Bonnet theorem

−tDD� ¼ ð M

by Theorem 3.5, where G�ðt;p;pÞ are the fundamental solutions of ∂<sup>t</sup> þ D�

<sup>∞</sup>ðt;p;pÞ ¼ <sup>1</sup>

ð4πtÞ

ð M

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

<sup>n</sup>=<sup>2</sup> ∑ ∞ k¼0

eH�

<sup>−</sup>λ<sup>k</sup> <sup>t</sup> <sup>¼</sup> <sup>1</sup> ð4πtÞ

ð4πtÞ

ð M ∑ ∞ k¼1 e

lim<sup>t</sup>!<sup>∞</sup> tr <sup>e</sup>

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>.

As far as Ind ðDÞ is concerned, it is the case for all t > 0 that,

<sup>−</sup>tD�<sup>D</sup>− tr e

G�ðt;p;pÞ

∑ k e

N a¼1

<sup>2</sup> <sup>¼</sup> <sup>∑</sup> ∞ k¼1

<sup>−</sup>tλ<sup>k</sup> 〈ψkðpÞ;ψkðpÞ〉

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

lim<sup>t</sup>!<sup>∞</sup> <sup>e</sup>

<sup>n</sup>=<sup>2</sup> <sup>þ</sup> <sup>O</sup> <sup>1</sup> t n=2 � �

tr Gþðt;p;pÞ dvp−

<sup>n</sup>=<sup>2</sup> ∑ ∞ m¼0 t ð M

mu�<sup>m</sup>ðp;p<sup>Þ</sup>

tr G−ðt;p;pÞ dvp

ukðx;xÞ dvx (52)

D and ∂<sup>t</sup> þ DD�

. As

∑ ∞ k¼1

<sup>e</sup><sup>−</sup>tλ<sup>k</sup> 〈ψkðpÞeaðpÞ〉ψkðpÞ;eaðp<sup>Þ</sup> � �

2

<sup>−</sup>tλ<sup>k</sup> <sup>¼</sup> tr <sup>ð</sup><sup>e</sup>

<sup>−</sup>tλ<sup>k</sup> <sup>¼</sup> dim <sup>Γ</sup>0ðE<sup>Þ</sup> (51)

−tΔÞ

<sup>e</sup><sup>−</sup>tλ<sup>k</sup> 〈ψkðpÞ;ψkðpÞ〉

<sup>2</sup> dvp <sup>¼</sup> <sup>∑</sup> ∞ k¼1 e

$$\sum\_{k} e^{-\lambda\_k t} = \int\_{M} \text{tr } G(t, \mathbf{x}, \mathbf{x}) \, d\upsilon\_{\mathbf{x}} = \frac{1}{(4\pi t)^{n/2}} \sum\_{k} \left( \int\_{M} \mu\_k(\mathbf{x}, \mathbf{x}) \, d\upsilon\_{\mathbf{x}} \right) t^{k/2}$$

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

$$\sum\_{i} \mathfrak{e}^{\lambda\_i t} \text{---} \frac{1}{(4\pi t)^{n/2}} \sum\_{k=0}^{\infty} a\_k \ t^k, \qquad a\_k = \int\_M u\_k(\mathfrak{x}, \mathfrak{x}) \, d\upsilon\_{\mathfrak{x}}$$

and the trace does not appear in the case of functions. The superscript on the Laplacian Δ<sup>p</sup> 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 ¼ m and dimN ¼ n. Then it follows that

$$\frac{1}{\left(\left(4\pi t\right)^{m/2}\right)\_{k=0}}\sum\_{k=0}^{\infty}\left(\int\_{M}u\_k^M(p,p)\,d\upsilon\_p\right)t^k = \sum\_{i=0}^{\infty}e^{-\lambda\_i t} = \frac{1}{\left(4\pi t\right)^{n/2}}\sum\_{k=0}^{\infty}\left(\int\_{N}u\_k^N(q,q)\,d\upsilon\_q\right)t^k$$

This implies that m ¼ n, which in turn implies that

$$\frac{1}{(4\pi t)^{m/2}} \left[ \int\_{M} u\_0^M(p,p) \, d\upsilon\_p - \int\_{N} u^N(q,q) \, d\upsilon\_q \right] = \frac{1}{(4\pi t)^{m/2}} \sum\_{k=1}^{m} \left( \int\_{M} u\_k^M(p,p) \, d\upsilon\_p - \int\_{N} u^N(q,q) \, d\upsilon\_q \right) t^k$$

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

$$\int\_{M} \mu\_0^M(p, p) \, d\upsilon\_p = \int\_N \mu\_0^N(q, q) \, d\upsilon\_q \tag{53}$$

Iterating this argument leads to the set of equations

$$\int\_{M} \mu\_k^M(p, p) \, d\upsilon\_p = \int\_{N} \mu\_k^N(q, q) \, d\upsilon\_q \tag{54}$$

for all k > 0. In particular, since u<sup>0</sup> ¼ 1, Eq. (53) leads to the conclusion vol ðMÞ ¼ 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 <sup>ð</sup> M uk dvp. 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 <sup>M</sup>. The notation <sup>P</sup>ðRp;∇Rp;…;∇<sup>k</sup> RpÞ 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,

$$\mu\_1(p, p) = P\_1^v(R\_p), \qquad \mu\_k(p, p) = P\_k^v(R\_p, \nabla R\_p, \dots, \nabla^{2k-2} R\_p), \qquad k \ge 2 \tag{55}$$

for some universal polynomials P<sup>n</sup> k .

Thus, P<sup>n</sup> <sup>1</sup> is a linear function with no constant term and u1ðp;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 u1ðp;pÞ on a manifold is the scalar curvature <sup>R</sup>ðpÞ ¼ Rij ij andso there exists a constant C such that u1ðp;pÞ ¼ C � RðpÞ.

Theorem 4.3.

$$
\mu\_1(p, p) = \frac{1}{6} R(p) \tag{56}
$$

Proof: The proof amounts to noticing that Pn <sup>1</sup> 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 u1ðp;pÞ ¼ nðn−1Þ=6 andit is known that <sup>R</sup>ðpÞ ¼ <sup>n</sup>ðn−1<sup>Þ</sup> for all <sup>p</sup>∈Sn 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

$$\sum\_{k=0}^{\infty} e^{-\lambda\_k t} = \frac{1}{(4\pi t)^{n/2}} \left\{ \upsilon(M) + \frac{1}{6} \int\_M R(\mathbf{x}) \, d\upsilon\_\mathbf{x} \cdot t + O(t^2) \right\}.$$

where vðMÞ is the volume of M.

Suppose that λ is positive and here we let Ep <sup>λ</sup> denote the possibly trivial eigenspace of Δ on pforms. If ω ∈ E<sup>p</sup> <sup>λ</sup> then it follows that <sup>Δ</sup><sup>p</sup>þ<sup>1</sup> <sup>d</sup><sup>ω</sup> <sup>¼</sup> <sup>d</sup>Δ<sup>p</sup> <sup>ω</sup> <sup>¼</sup> <sup>λ</sup> <sup>d</sup>ω, hence <sup>d</sup><sup>ω</sup> <sup>∈</sup> <sup>E</sup><sup>p</sup>þ<sup>1</sup> <sup>λ</sup> . Thus, a welldefined sequential ordering of the spaces can be established. If ω∈E<sup>p</sup> <sup>λ</sup> has the property that <sup>d</sup><sup>ω</sup> <sup>¼</sup> 0, then λω <sup>¼</sup> <sup>Δ</sup><sup>p</sup> <sup>ω</sup> ¼ ðδ<sup>d</sup> <sup>þ</sup> <sup>d</sup>δÞ<sup>ω</sup> <sup>¼</sup> <sup>d</sup> δω. Therefore, since <sup>λ</sup>≠0, it is found that <sup>ω</sup> <sup>¼</sup> <sup>d</sup> <sup>1</sup> <sup>λ</sup> δω � �. Thus, the sequence 0 ! <sup>E</sup><sup>0</sup> <sup>λ</sup> !<sup>d</sup> <sup>⋯</sup>!<sup>d</sup> <sup>E</sup><sup>n</sup> <sup>λ</sup> ! 0 is exact. Since the operator d þ δ is an isomorphism on ⊕<sup>k</sup> E<sup>2</sup><sup>k</sup> <sup>λ</sup> , it follows that

$$\sum\_{s} (-1)^{s} \dim E\_{\lambda}^{s} = 0 \tag{57}$$

Theorem 4.4. Let {λ<sup>s</sup> <sup>i</sup>} be the spectrum of the operator Δ, then

$$\sum\_{s} (-1)^{s} \sum\_{i} e^{-\lambda\_{i}^{s} t} = \sum\_{s} (-1)^{s} \text{dim } \text{ker } \Delta^{s}. \tag{58}$$

Proof: By (57),

The proof illustrates that in fact there exist an infinite sequence of obstructions to claiming that

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

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

ð M

<sup>k</sup> <sup>ð</sup>Rp;∇Rp;…;∇<sup>2</sup>k−<sup>2</sup>

<sup>1</sup> is a linear function with no constant term and u1ðp;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 u1ðp;pÞ on a

<sup>u</sup>1ðp;pÞ ¼ <sup>1</sup>

6

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 u1ðp;pÞ ¼ nðn−1Þ=6

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

<sup>n</sup>=<sup>2</sup> <sup>v</sup>ðMÞ þ <sup>1</sup>

6 ð M

RðxÞ dvx � t þ Oðt

� �

uk dvp. The first integral contains

RpÞ is used to denote a

RpÞ; k ≥ 2 (55)

ij andso there exists a constant C such that

RðpÞ (56)

<sup>1</sup> is a universal polynomial, so it suffices to

2 Þ

two manifolds are isospectral, namely the set of integrals

Theorem 4.2. On a manifold of dimension n,

<sup>u</sup>1ðp; <sup>p</sup>Þ ¼ Pn

manifold is the scalar curvature <sup>R</sup>ðpÞ ¼ Rij

Proof: The proof amounts to noticing that Pn

this theorem will be developed here.

where vðMÞ is the volume of M.

∑ ∞ k¼0 e

These results can be summarized by the elegant formula

<sup>−</sup>λ<sup>k</sup> <sup>t</sup> <sup>¼</sup> <sup>1</sup> ð4πtÞ

for some universal polynomials P<sup>n</sup>

will be used shortly.

84 Manifolds - Current Research Areas

Thus, P<sup>n</sup>

u1ðp;pÞ ¼ C � RðpÞ.

Theorem 4.3.

depend only on the dimension of <sup>M</sup>. The notation <sup>P</sup>ðRp;∇Rp;…;∇<sup>k</sup>

<sup>1</sup> <sup>ð</sup>RpÞ; ukðp;pÞ ¼ <sup>P</sup><sup>n</sup>

k .

andit is known that <sup>R</sup>ðpÞ ¼ <sup>n</sup>ðn−1<sup>Þ</sup> for all <sup>p</sup>∈Sn andthis implies Eq. (56).

$$\sum\_{s}(-1)^{s}\sum\_{k}e^{-\lambda\_{k}^{s}t} = \sum\_{s}(-1)^{s}\sum\_{i}e^{-\lambda\_{i}t}$$

The sum on the right ∑′ is only over eigenvalues such that λ<sup>p</sup> <sup>i</sup> ¼ 0 and so

$$\sum^{'} e^{-\lambda\_i^p t} = \dim \ker \Delta^p.$$

This has the consequence that

$$\sum\_{p} (-1)^{p} \text{ tr } e^{-t\Lambda} = \sum\_{p} (-1)^{p} \sum\_{k} e^{-\lambda\_{k}^{\overline{r}}t} \tag{59}$$

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 tr e<sup>−</sup>t<sup>Δ</sup> 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

$$\begin{split} \chi(M) = \sum\_{\mathbb{P}} (-1)^{\mathbb{P}} \dim H^{\mathbb{P}}\_{\text{dfl}}(M) &= \sum\_{\mathbb{P}} (-1)^{\mathbb{P}} \dim \ker \Delta^{\mathbb{P}} = \sum\_{\mathbb{P}} (-1)^{\mathbb{P}} \text{ tr } e^{-t\Delta^{\mathbb{P}}} \\ &= \sum\_{\mathbb{P}} (-1)^{\mathbb{P}} \int\_{M} \text{ tr } G(t, \mathbf{x}, \mathbf{x}) \, d\upsilon\_{\mathbf{x}} \end{split} \tag{60}$$

From the asymptotic expansion theorem, the following expression for χðMÞ results

$$\chi(M) = \frac{1}{\left(4\pi t\right)^{n/2}} \sum\_{k=0}^{\infty} \left(\int\_{M} \sum\_{s=0}^{n} \text{tr } \boldsymbol{u}\_{k}^{s}(\mathbf{x}, \mathbf{x}) \, d\boldsymbol{v}\_{k}\right) t^{k} \tag{61}$$

The us <sup>k</sup> in Eq. (61) are the coefficients in the asymptotic expansion for tr <sup>ð</sup>e<sup>−</sup>tΔ<sup>s</sup> Þ. Since χðMÞ 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

$$\frac{1}{\left(4\pi\right)^{n/2}} \int\_{M} \sum\_{s=0}^{n} \left(-1\right)^{s} \text{ tr } u\_{k}^{s}(\mathbf{x}, \mathbf{x}) \, dv\_{\mathbf{x}} = \begin{cases} 0, & k\theta \frac{n}{2};\\ \chi(M), & k = \frac{n}{2}. \end{cases} \tag{62}$$

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

$$\chi(M) = \frac{1}{2\pi} \int\_{M} K \, da\_{M} \tag{63}$$

Proof: By the last theorem and the fact that tr up <sup>k</sup> <sup>ð</sup>x;xÞ ¼ tr <sup>u</sup><sup>p</sup>−<sup>1</sup> <sup>k</sup> ðx;xÞ, it follows that

$$\begin{aligned} \chi(M) &= \frac{1}{4\pi} \int\_{M=0} \sum\_{M=0}^{2} \begin{pmatrix} -^{1} \mathbb{I} & \text{tr } u\_{1}^{p} \, da\_{M} = \frac{1}{4\pi} \int\_{M} \left( \text{tr } u\_{1}^{0} - \text{tr } u\_{1}^{1} + \text{tr } u\_{1}^{2} \right) \, da\_{M} \\\\ &= \frac{1}{4\pi} \int\_{M} \begin{pmatrix} 2 \ \text{tr } u\_{1}^{0} - \text{tr } u\_{1}^{1} \end{pmatrix} \, da\_{M} = \frac{1}{4\pi} \int\_{M} \begin{pmatrix} 2 \ \text{K} - \text{tr } u\_{1}^{1} \end{pmatrix} \, da\_{M} \end{aligned} \tag{64}$$

since the scalar curvature is two times the Gaussian. Now it must be that tr u<sup>1</sup> <sup>1</sup>ðx;xÞ ¼ CRðxÞ ¼ <sup>2</sup>CKðxÞ, for some constant <sup>C</sup>. The standard sphere <sup>S</sup><sup>2</sup> has Gaussian curvature one andso C can be calculated from Eq. (64),

$$\mathcal{Q} = \frac{1}{2\pi} \int\_{S^2} (\frac{1}{3} - \mathcal{C}) \, da\_M = \frac{1}{2\pi} (\frac{1}{3} - \mathcal{C}) \cdot (4\pi)^2$$

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<sup>1</sup> gives another topological obstruction to manifolds having the same spectrum.

Theorem 4.7. Let ðM;gÞ and ðN;hÞ be compact isospectral surfaces, then M and N are diffeomorphic.

Proof: As noted in Corollary 4.1,

$$\int\_{M} u\_1^{\mathcal{M}}(\mathfrak{x}, \mathfrak{x}) \, dv\_{\mathfrak{x}} = \int\_{N} u\_1^{\mathcal{N}}(\mathfrak{y}, \mathfrak{y}) \, dv\_{\mathfrak{y}}$$

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

$$\left(\mathfrak{G}\pi\chi(M) = \int\_{M} u\_1^M(\mathbf{x}, \mathbf{x}) \, dv\_{\mathbf{x}} = \int\_{N} u\_1^N(y, y) \, dv\_y = \mathfrak{G}\pi\chi(\mathbf{N})\tag{65}$$

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