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

The generalized-Gauss-Bonnet theorem states that

[itex]\int_M \mbox{Pf}(\Omega)=2^n\pi^n\chi(M)[itex]

where [itex]\chi(M)[itex] 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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