2011/07/08 by Bruno Mendonça, Ruy Tojeiro, Mendonça, Bruno +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG
paper · pdf · doi:10.48550/arxiv.1107.1679
28 pages, two corollaries added, several corrections made
arxiv created 2011/08/26 · arxiv updated 2011/08/29
We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of \Sfn× \R, extending the classification of umbilical surfaces in \Sf2× \R by Rabah-Souam and Toubiana as well as the local description of umbilical hypersurfaces in \Sfn× \R by Van der Veken and Vrancken. We prove that, besides small spheres in a slice, up to isometries of the ambient space they come in a two-parameter family of rotational submanifolds whose substantial codimension is either one or two and whose profile is a curve in a totally geodesic \Sf1× \R or \Sf2× \R, respectively, the former case arising in a one-parameter family. All of them are diffeomorphic to a sphere, except for a single element that is diffeomorphic to Euclidean space. We obtain explicit parametrizations of all such submanifolds. We also study more general classes of submanifolds of \Sfn× \R and \Hyn× \R. In particular, we give a complete description of all submanifolds in those product spaces for which the tangent component of a unit vector field spanning the factor \R is an eigenvector of all shape operators. We show that surfaces with parallel mean curvature vector in \Sfn× \R and \Hyn× \R having this property are rotational surfaces. We also prove a Dajczer-type reduction of codimension theorem for submanifolds of \Sfn× \R and \Hyn× \R.