Generalized Gauss-Bonnet theorem
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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 <math>\Omega<math> be the curvature form of the Levi-Civita connection. This means that <math>\Omega<math> is an <math>\mathfrak s\mathfrak o(2n)<math>-valued 2-form on M. So <math>\Omega<math> can be regarded as a skew-symmetric 2n × 2n matrix whose entries are 2-forms, so it is a matrix over the commutative ring <math>\bigwedge^{\hbox{even}}T^*M<math>. One may therefore take the Pfaffian of <math>\Omega<math>, <math>\mbox{Pf}(\Omega)<math> which turns out to be a 2n-form.
The generalized-Gauss-Bonnet theorem states that
- <math>\int_M \mbox{Pf}(\Omega)=2^n\pi^n\chi(M)<math>
where <math>\chi(M)<math> denotes the Euler characteristic of M.
Further generalizations
As with the Gauss-Bonnet theorem, there are generalizations when M is a manifold with boundary.