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Spacelike submanifolds, their umbilical properties and applications to gravitational physics.

We give a characterization theorem for umbilical spacelike submanifolds of arbitrary dimension and co-dimension immersed in a semi-Riemannian manifold. Letting the codimension arbitrary implies that the submanifold may be umbilical with respect to some subset of normal directions. This leads to the definition of umbilical space and to the study of its dimension. The trace-free part of the second fundamental form, called total shear tensor in this thesis, plays a central role in the characterization theorems. It allows us to define shear objects (shear operators, shear tensors and shear scalars) that determine the umbilical properties of the spacelike submanifold with respect to a given normal vector field. Given a group of conformal motions G acting on a semi-Riemannian manifold and an orbit S, we apply the characterization results in order to find necessary and sufficient conditions for S to have a non-empty umbilical space. We prove that if the isotropy subgroup of G is trivial, then the umbilical condition depends on the scalar products of a set of generating conformal Killing vector fields. If the isotropy subgroup of G is non-trivial, we argue that, under specific assumptions, it is possible to prove that the umbilical condition is automatically satisfied so that the umbilical space is non-trivial. The assumptions would depend on the co-dimension of S, the dimension of the isotropy subgroup and the ranks of specific matrices defined in terms of the structure constants of G. In the last part of the thesis we consider Lorentzian warped productsM = M ×f Y and we analyse a particular class of spacelike submanifolds S. We find a sufficient condition that allows us to prove, on one hand, the existence of focal points along timelike or null geodesics normal to S and, on the other hand, the null geodesic incompleteness ofM under additional reasonable conditions. By assuming that we can split the immersion as S → Σ→M, where Σ is eitherM×{q} or {q}×Y , we find that the Galloway-Senovilla condition [29] can be written in terms of the warping function f and the Riemann tensor of either only M or Y . This means that, for instance, in order to prove singularity theorems one can restrict the study to just one of the two manifolds defining the warped product rather than considering the warped product manifold itself. We translate the condition found to some specific situations, such as positive and constant sectional curvature, Einstein and Ricci-flat spaces and to a few subcases in terms of the co-dimension of S. The same has been done in direct products (f = 1).