Generalized Gauss–Bonnet theorem

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In mathematics, the generalized Gauss–Bonnet theorem (also called Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss and Pierre Ossian Bonnet) presents the Euler characteristic of a closed even-dimensional Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of the Gauss–Bonnet theorem (named after Carl Friedrich Gauss and Pierre Ossian Bonnet) to higher dimensions.

Let M be a compact orientable 2n-dimensional Riemannian manifold without boundary, and let ${\displaystyle \Omega }$ be the curvature form of the Levi-Civita connection. This means that ${\displaystyle \Omega }$ is an ${\displaystyle {\mathfrak {s}}{\mathfrak {o}}(2n)}$-valued 2-form on M. So ${\displaystyle \Omega }$ can be regarded as a skew-symmetric 2n × 2n matrix whose entries are 2-forms, so it is a matrix over the commutative ring ${\displaystyle \wedge ^{\hbox{even}}\,T^{*}M}$. One may therefore take the Pfaffian ${\displaystyle {\mbox{Pf}}(\Omega )}$, which turns out to be a 2n-form.

The generalized Gauss–Bonnet theorem states that

${\displaystyle \int _{M}{\mbox{Pf}}(\Omega )=(2\pi )^{n}\chi (M)\ }$

where ${\displaystyle \chi (M)}$ denotes the Euler characteristic of M.

Example: four dimensional manifolds

In dimension ${\displaystyle 2n=4}$, for a compact oriented manifold, we get

${\displaystyle \chi (M)={\frac {1}{32\pi ^{2}}}\int _{M}\left(|{\text{Riem}}|^{2}-4|{\text{Ric}}|^{2}+R^{2}\right)d\mu }$

where ${\displaystyle {\text{Riem}}}$ is the full Riemann curvature tensor, ${\displaystyle {\text{Ric}}}$ is the Ricci curvature tensor, and ${\displaystyle R}$ is the scalar curvature.

Further generalizations

As with the two-dimensional Gauss–Bonnet theorem, there are generalizations when M is a manifold with boundary.

The Gauss–Bonnet theorem can be seen as a special instance in the theory of characteristic classes. The Gauss–Bonnet integrand is the Euler class. Since it is a top-dimensional differential form, it is closed. The naturality of the Euler class means that when you change the Riemannian metric, you stay in the same cohomology class. That means that the integral of the Euler class remains constant as you vary the metric, and so is an invariant of smooth structure.

A far-reaching generalization of the Gauss–Bonnet theorem is the Atiyah–Singer Index Theorem. Let ${\displaystyle D}$ be a (weakly) elliptic differential operator between vector bundles. That means that the principal symbol is an isomorphism. (Strong ellipticity would furthermore require the symbol to be positive-definite.) Let ${\displaystyle D^{*}}$ be the adjoint operator. Then the index is defined as dim(ker(D))-dim(ker(D*)), and by ellipticity is always finite. The Index Theorem states that this analytical index is constant as you vary the elliptic operator smoothly. It is in fact equal to a topological index, which can be expressed in terms of characteristic classes. The 2-dimensional Gauss–Bonnet theorem arises as the special case where the topological index is defined in terms of Betti numbers and the analytical index is defined in terms of the Gauss–Bonnet integrand.