Umbilical Submanifolds of S-n x R

Author(s)

Mendonca, Bruno ^[1] ; Tojeiro, Ruy ^[2]

Affiliation

^[1] Univ Estadual Londrina, BR-86051980 Londrina - Brazil

^[2] Univ Fed Sao Carlos, BR-13565905 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES; v. 66, n. 2, p. 400-428, APR 2014.
• Web of Science Citations: 9

Abstract

We give a complete classification of umbilical submanifolds of arbitrary dimension and codimension of S-n x R, extending the classification of umbilical surfaces in s(2) x R by Souam and Toubiana as well as the local description of umbilical hypersurfaces in S-n x 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 51 x Ill or s(2) x 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 S-n x R and H-n x 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 S-n x R and H-n x R having this property are rotational surfaces, and use this fact to improve some recent results by Alencar, do Carmo, and Tribuzy. We also obtain a Dajczer-type reduction of codimension theorem for submanifolds of S-n x R and H-n x R. (AU)

• FAPESP's process: 11/21362-2 - Group actions, submanifold theory and global analysis in Riemannian and pseudo-Riemannian geometry
• Grantee: Paolo Piccione
• Support Opportunities: Research Projects - Thematic Grants