On spacelike immersions in locally symmetric semi-Riemannian spaces.

Abstract

In the first part of this these we study the geometry of immersions of the spacelike hypersurfaces in constant sectional curvature space, more specifically into the Steady State Space Hn+1 1 and Anti-De Sitter space Hn+1 1 . In these results, we use suitable conditions on the behavior of higher order mean curvatures Hr to prove some results of characterizations of totally umbilical hypersurfaces in the Hn+1 and Hn+1 1 , also in this process was use an suitable extension of the Omori-Yau’s generalized maximum principle due to Alías, Impera and Rigoli in [10]. In the second part we study the geometry of spacelike submanifolds with parallel normalized mean curvature vector in constant sectional curvature spaces, where we use polynomial volume growth techniques and a maximum principle at infinity established by Alías, Caminha and Nascimento [7], our objects have hypotheses like: stochastically completeness, L parabolicity and L1-Lioville to ensure that a given submanifold is totally umbilical. In the third and last part, we study the geometry of linear Weingarten spacelike complete submanifolds immersed with parallel normalized mean curvature vector and flat normal bundle in locally symmetric semi-Riemannian spaces Ln+p p with index p. In this sense, our ob jective was to establish sufficient conditions to guarantee that a given submanifold Mn is totally umbilical or isometric to an isoparametric hypersurface of a totally geodesic submanifold Ln+1 1 ,! Ln+p p .