Affine differential geometry of the unit normal vector fields of hypersurfaces in the real space forms

Abstract

In this paper, for a hypersurface in the real space form of constant curvature, we prove that the unit normal vector field is an affine imbedding into a certain sphere bundle with canonical metric. Moreover, we study the relations between a hypersurface and its unit normal vector field as an affine imbedding. In particular, several hypersurfaces are characterized by affine geometric conditions which are independent of the choice of the transversal bundle.