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

\Omega

be the curvature form of the Levi-Civita connection. This means that

\Omega

is an

\mathfrak s\mathfrak o(2n)

-valued 2-form on M. So

\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

\bigwedge^{\hbox{even}}T^*M

. One may therefore take the Pfaffian of

\Omega

\mbox{Pf}(\Omega)

which turns out to be a 2n-form. The generalized-Gauss-Bonnet theorem states that

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

where

\chi(M)

denotes the Euler characteristic of M.

Further generalizations

As with the Gauss-Bonnet theorem, there are generalizations when M is a manifold with boundary.

Generalized Gauss-bonnet Theorem

Further generalizations

See also: