On the volume of the generalized Gauss map

Abstract

Let f :M~ N"+P be an immersion of the n-manifold M into Euclidean (n + p)-space. The classical Gauss map for surfaces in 3-space has two natural generalizations to this situation. In the obvious way we can assign (a) to every unit normal vector of f a point in the unit sphere S "+p- 1, or (b) to every point of M its p-dimensional normal space through tile origin of ~"+P. Both the sphere S "+p- 1 and the Grassmannian Gv(~ "+p) have canonical Riemannian metrics, and we can talk about the (n + p - 1)-dimensional or the n-dimensional volumes of the Gauss maps (a) and (b). In the first case this (suitably normalized) volume is the well-known total absolute curvature r(f) off. The Gauss map (b) has also been studied extensively (see [8] for a survey and a list of references). In the present note we shall affirmatively answer a conjecture of Osserman [8] stating that the (again suitably normalized) volume a(f) of the map (b) satisfies z(f) 0 denote the volume of the unit k-sphere, and define