5. Summary and outlook

Theorem 4.5. If the dimension of M is even, then

1 ð4πÞ n=2 ð M ∑ n s¼0 ð−1Þ

Proof: By the last theorem and the fact that tr up

4π ð M ∑ 2 p¼0 ð

¼ 1 4π ð M

andso C can be calculated from Eq. (64),

diffeomorphic.

theorem,

Proof: As noted in Corollary 4.1,

<sup>χ</sup>ðMÞ ¼ <sup>1</sup>

and area measure daM, then

86 Manifolds - Current Research Areas

<sup>s</sup> tr us

<sup>χ</sup>ðMÞ ¼ <sup>1</sup>

<sup>−</sup>1Þ<sup>p</sup> tr up

<sup>1</sup>− tr u<sup>1</sup>

Therefore, C ¼ −2=3 and putting all of these results into Eq. (64), Eq. (62) results.

<sup>1</sup> ðx;xÞ dvx ¼

<sup>1</sup> ðx;xÞ dvx ¼

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

<sup>ð</sup>2 tr <sup>u</sup><sup>0</sup>

<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2π ð S2 ð 1 3

> ð M uM

ð M u<sup>M</sup>

obstruction to manifolds having the same spectrum.

6πχðMÞ ¼

Theorem 4.6. (Gauss-Bonnet) Let M be a closed oriented manifold with Gaussian curvature K

2π ð M

<sup>1</sup> daM <sup>¼</sup> <sup>1</sup>

since the scalar curvature is two times the Gaussian. Now it must be that tr u<sup>1</sup>

4π ð M <sup>ð</sup> tr <sup>u</sup><sup>0</sup>

<sup>1</sup><sup>Þ</sup> daM <sup>¼</sup> <sup>1</sup>

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

<sup>−</sup>C<sup>Þ</sup> daM <sup>¼</sup> <sup>1</sup>

As an application of this theorem, note that the calculation of u<sup>1</sup> gives another topological

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

ð N uN

ð N uN

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

<sup>1</sup> ðy;yÞ dvy

4π ð M ð 2 3

2π ð 1 3

<sup>k</sup>ðx;x<sup>Þ</sup> dvx <sup>¼</sup> <sup>0</sup>; <sup>k</sup><sup>≠</sup>

<sup>k</sup> <sup>ð</sup>x;xÞ ¼ tr <sup>u</sup><sup>p</sup>−<sup>1</sup>

<sup>1</sup>− tr u<sup>1</sup>

−CÞ�ð4πÞ

K− tr u<sup>1</sup>

8 < : n 2 ;

2 :

K daM (63)

<sup>k</sup> ðx;xÞ, it follows that

<sup>1</sup>Þ daM

<sup>1</sup> <sup>þ</sup> tr <sup>u</sup><sup>2</sup>

<sup>1</sup>Þ daM

<sup>1</sup> ðy;yÞ dvy ¼ 6πχðNÞ (65)

(62)

(64)

<sup>1</sup>ðx;xÞ ¼

<sup>χ</sup>ðMÞ; <sup>k</sup> <sup>¼</sup> <sup>n</sup>

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 <sup>1</sup> <sup>32</sup>π<sup>2</sup> {K<sup>2</sup> −4jρrj <sup>2</sup> þ jR<sup>j</sup> 2 }, where K is the scalar curvature,jρrj <sup>2</sup> is the norm of theRicci tensor,jR<sup>j</sup> <sup>2</sup> 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 thisinvariantis used to get the associated Euler-Lagrange equations.

Let Rijkl be the components of the Riemann curvature tensor relative to an arbitrary local frame field {ei} for the tangent bundle TM and adopt the Einstein summation convention. Let m ¼ 2s be even, then the Pfaffian EmðgÞ is defined to be

$$E\_m(\mathbf{g}) = \frac{1}{(8\pi)^s s!} \ R\_{i\_1 i\_2 j\_1} \dotsm R\_{i\_{2s-1} i\_{2j\_2 j\_{2s}}} \operatorname{g} (\boldsymbol{\varepsilon}^{j\_1} \wedge \dotsm \wedge \boldsymbol{\varepsilon}^{j\_{2s}}, \boldsymbol{\varepsilon}^{j\_1} \wedge \dotsm \wedge \boldsymbol{\varepsilon}^{j\_{2s}}) \tag{66}$$

The Euler characteristic χðMÞ of any compact manifold of odd dimension without boundary vanishes. Only the even dimensional case is of interest.

Theorem 5.1. Let ðM;gÞ be a compact Riemannian manifold without boundary of even dimension m. Then

$$\chi(M) = \int\_M E\_m(\mathcal{g}) \, d\upsilon\_M \tag{67}$$

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.
