The Gauss–Bonnet Formula for Hyperbolic Manifolds of Finite Volume

Let M denote an even-dimensional noncompact hyperbolic manifold of finite volume. We show that such manifolds are candidates for minimal volume. Generalizing H. Hopf's ideas around the Curvatura integra for compact Clifford–Klein space forms, we present an elementary combinatorial-metrical proof of the Gauss–Bonnet formula for M. In contrast to former results of G. Harder and M. Gromov, our approach doesn't make use of the arithmetical and differential geometrical machinery.