Sobre a geometria de imersões isométricas no espaço hiperbólico com aplicação de Gauss prescrita.

Abstract

In this work we study the geometry of a submanifold Mn, n 2, isometrically immersed in the hyperbolic space, Hn+p, p 1, with some prescribed conditions on the behavior of its Gauss application. In the case p = 1, initially our goal is to show that a complete hypersurface Mn with constant mean curvature is totally umbilical, provided that N(Mn) lies in a totally umbilical spacelike hypersurface of the de Sitter space Sn+11 . Next, we show another result for the same conclusion but this time we assume that Mn has scalar curvature bounded from below and that N(Mn) is contained in a certain region of Sn+1 1 determined by some vector a of the Lorentz-Minkowski space Ln+2. Finally, in the case p > 1 we establish su cient conditions to guarantee a complete submanifolds Mn with parallel nonzero mean curvature vector must be pseudo-umbilical. In particular, we conclude that Mn is a minimal submanifold of a small hypersphere of Hn+p.