Generalized Gauss-Bonnet theorem

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In mathematics, the generalized-Gauss-Bonnet theorem 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 to higher dimensions.

Let M be a compact 2n-dimensional Riemannian manifold without boundary, and let $Ω$ 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.