THE GENERALIZED GAUSS-BONNET-CHERN THEOREM

For Riemannian manifolds with boundary, the well-known Gauss-Bonnet-Chern theorem gives an integral formula for the Euler characteristic of the manifold. Here we extend a proof by Avez to show that there is a similar result for manifolds with boundary endowed with a pseudo-Riemannian metric of arbitrary signature. In the case when the metric is Lorentzian there are some applications to general relativity. The generalized Gauss-Bonnet-Chern theorem also provides a formula for the gravitational kink. © 1995 American Institute of Physics.