Differential topology and the computation of total absolute curvature

Abstract

The total absolute (Lipschitz-Killing-)curvature of an immersion of a manifold in Euclidean space is equal to the mean value of the number of critical points of certain level functions which are induced by bundles of parallel hyperplanes (see Chern-Lashof and Kuiper). We generalize this for immersions in spaces of constant curvature + 1. A level function is induced by an oriented hyperplane bundle either through a common plane of codimension two or orthogonal to a geodesic. In the hyperbolic case we have to restrict the level function to the interior of the normal tube of constant radius around that plane or geodesic.