Generalized Gauss–Bonnet theorem

In mathematics, the generalized Gauss–Bonnet theorem (also called Chern–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 (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.

...
Wikipedia