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Generalized Gauss-Bonnet theorem

(Redirected from Chern-Gauss-Bonnet theorem)

In mathematics, the generalized-Gauss-Bonnet theorem presents the Euler characteristic of a closed Riemannian manifold as an integral of a certain polynomial derived from its curvature. It is a direct generalization of the Gauss-Bonnet theorem to general even dimension.

Let M be a compact Riemannian manifold of dimension 2n and $Ω$ be the curvature form of the Levi-Civita connection. This means that $Ω$ is an $\mathfrak s\mathfrak o(2n)$ -valued 2-form on M. So $Ω$ can be regarded as a skew-symmetric 2n × 2n matrix whose entries are 2-forms, so it is a matrix over the commutative ring $\bigwedge^{\hbox{even}}T^*M$ . One may therefore take the Pfaffian of $Ω$ , $Pf(Ω)$ which turns out to be a 2n-form.

The generalized-Gauss-Bonnet theorem states that

\int Pf(Ω) = 2 n π n χ(M) M

where $χ(M)$ denotes the Euler characteristic of M.

Further generalizations

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

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