Pseudo-parallel immersions of Lorentzian manifolds in pseudo-Riemannian space forms

Abstract

In this Ph.D. thesis, we study pseudo-parallel submanifolds in pseudo-Riemannian space forms. We give a characterization of pseudo-parallel Lorentzian surfaces with non-flat normal bundle in pseudo-Riemannian space forms as λ-isotropic surfaces, extending an analogous result by Asperti-Lobos-Mercuri in the Riemannian case. Consequently, for this kind of Lorentzian surfaces we give a characterization using the concept of hyperbola of curvature and get a non-existence result when the ambient space is a Lorentzian space form. In particular, when the ambient space is a 4-dimensional pseudo-Riemannian space form, we obtain that any pseudo-parallel Lorentzian surface with non-flat normal bundle is super-extremal, i.e., a λ-isotropic surface with everywhere vanishing mean curvature vector field, and the ambient space must have metric of index 2. In the case where the pseudo-parallelism function is constant, we explicitly describe these surfaces with codimension two, obtaining that they are parallel surfaces and exist in non-flat space forms, and for the case where the pseudo-parallelism function is non-constant we give explicit examples of these surfaces in the 4-dimensional pseudo-Euclidean space with metric of index 2. An example of an extremal and flat pseudo-parallel Lorentzian surface with non-flat normal bundle which is not semi-parallel is given in codimension three. We continue the study of pseudo-parallel Lorentzian hypersurfaces in Lorentzian space forms started by Lobos, by completing the characterization of the Weingarten operator even when it is non-diagonalizable. Then, we consider the case where the pseudo-parallelism function is constant and different from the curvature of the ambient space and give the local classification of these hypersurfaces under the hypothesis of being good in the sense of Ryan. We also give a classification of the connected complete semi-parallel Lorentzian hypersurfaces of the Minkowski space and a local classification of the pseudo-parallel Lorentzian hypersurfaces with constant pseudo-parallelism function and constant mean curvature in Lorentzian space forms.