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A Simons Type Formula for Spacelike Submanifolds in Semi-Riemannian Warped Product and its Applications

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Autor(es):
Lobos, Guillermo A. ; Melara, Mynor ; Santos, Maria R. B.
Número total de Autores: 3
Tipo de documento: Artigo Científico
Fonte: Results in Mathematics; v. 79, n. 8, p. 22-pg., 2024-12-01.
Resumo

We determine a Simons type formula for spacelike submanifolds within a broad class of semi-Riemannian warped products, which includes the Robertson-Walker spacetimes. This formula enables us to extend results well-known from Riemannian geometry to the semi-Riemannian case. Specifically, when the ambient space has constant sectional curvature, such as Lorentz-Minkowski, de Sitter and anti-de Sitter spacetimes, we establish that compact spacelike hypersurfaces with parallel mean curvature vector field and non-negative sectional curvature are isoparametric hypersurfaces. This result constitutes a generalization of the Riemannian case within space forms, as demonstrated by Nomizu and Smyth in 1969. As an application, we extend analogous results previously established for pseudo-parallel immersions in Riemannian space forms. With this extension, we prove that any semi-parallel spacelike hypersurface with zero mean curvature into de Sitter spacetime is totally geodesic. Furthermore, we show that there are no semi-parallel spacelike hypersurfaces with zero mean curvature into Einstein-de Sitter spacetime. (AU)

Processo FAPESP: 22/16097-2 - Métodos modernos em geometria diferencial e análise geométrica
Beneficiário:Paolo Piccione
Modalidade de apoio: Auxílio à Pesquisa - Temático