Sobre a geometria de imersões Riemannianas em variedades Semi-Riemannianas.

Abstract

In this thesis, we initially established a characterization theorem concerning to complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, whose sectional curvature is supposed to obey certain appropriated conditions. Under a suitable restriction on the length of the second fundamental form, we prove that a such spacelike hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple. Afterwards, we obtain the same result, when the locally symmetric Lorentz space is Einstein, by using as main analytical tool a generalized maximum principle for complete noncompact Riemannian manifolds. Following, we study the uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm and such that its fiber has constant sectional curvature. By using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds and supposing a natural comparison inequality between the r-th mean curvatures of the hypersurface and that ones of the slices of the region where the hypersurface is contained, we are able to prove that a such hypersurface must be, in fact, a slice. Finally, we study the geometry of entire conformal Killing graphs, that is, graphs constructed through the flow generated by a complete conformal Killing vector field V and which are defined over an integral leaf of the foliation V⊥ orthogonal to V. In this setting, under a suitable restriction on the norm of the gradient of the function z which determines such a graph ∑(z), we establish sufficient conditions to ensure that ∑(z) is totally umbilical and, in particular, an integral leaf of V⊥. We too establish sufficient conditions to ensure that ∑(z) is totally geodesic. Afterwards, when the ambient space M has constant sectional curvature, we obtain lower estimates for the index of minimum relative nullity of ∑(z).