Homogeneous hypersurfaces and totally geodesic submanifolds

Abstract

This Ph.D. thesis deals with the study of certain classes of submanifolds in the presence of symmetry. Namely, results have been derived regarding the theory of submanifolds in Riemannian homogeneous spaces with a special emphasis on symmetric spaces. In this dissertation, we will focus on two of the most natural classes of submanifolds that one can study in Riemannian manifolds. These are homogeneous hypersurfaces and totally geodesic submanifolds. Regarding the first ones, we will conclude the classification of homogeneous hypersurfaces in symmetric spaces of rank one, by finishing the classification in quaternionic hyperbolic spaces. As for totally geodesic submanifolds, we will derive different classifications. In particular, we will classify totally geodesic submanifolds in the following spaces: in products of symmetric spaces of rank one, in exceptional symmetric spaces, and in Hopf-Berger spheres.