Isometric immersions with homothetical Gauss map

Abstract

Let f:M→ℝ be an isometric immersion of an m-dimensional Riemannian manifold M into the n-dimensional Euclidean space. Its Gauss map g:M→G m (ℝ n ) into the Grassmannian G m (ℝ n ) is defined by assigning to every point of M its tangent space, considered as a vector subspace of ℝ n . The third fundamental form b of f is the pull-back of the canonical Riemannian metric on G m (ℝ n ) via g. In this article we derive a complete classification of all those f (with flat normal bundle) for which the Gauss map g is homothetical; i.e. b is a constant multiple of the Riemannian metric on M. Using these results we furthermore classify all those f (with flat normal bundle) for which the third fundamental form b is parallel w.r.t. the Levi-Civita connection on M.