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Generalized Gauss-Bonnet theorem
From Wikipedia, the free encyclopedia
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 
-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 
. 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
| ∫ | Pf(Ω) = (2π)nχ(M) |
| M |
where χ(M) denotes the Euler characteristic of M.
As with the Gauss-Bonnet theorem, there are generalizations when M is a manifold with boundary.