| Texto completo | |
| Autor(es): |
Número total de Autores: 2
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| Afiliação do(s) autor(es): | [1] Univ Estadual Londrina, BR-86051980 Londrina - Brazil
[2] Univ Fed Sao Carlos, BR-13565905 Sao Carlos, SP - Brazil
Número total de Afiliações: 2
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| Tipo de documento: | Artigo Científico |
| Fonte: | CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES; v. 66, n. 2, p. 400-428, APR 2014. |
| Citações Web of Science: | 9 |
| Resumo | |
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) | |
| Processo FAPESP: | 11/21362-2 - Ações de grupos, teoria de subvariedades, e análise global em geometria Riemanniana e pseudo-riemanniana |
| Beneficiário: | Paolo Piccione |
| Modalidade de apoio: | Auxílio à Pesquisa - Temático |