LIMA, J. R.; http://lattes.cnpq.br/2905593074062857; LIMA, Joseilson Raimundo de.
Resumen:
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).