Homogeneous hypersurfaces in symmetric spaces

Abstract

A hypersurface of a Riemannian manifold is said to be (extrinsically) homogeneous if it can be obtained as an orbit of an action of a subgroup of the isometry group of the ambient space. In this case, such an action is said to be of cohomogeneity one. The study of homogeneous hypersurfaces only makes sense for ambient spaces with a large enough isometry group. This is the case of Riemannian symmetric spaces, which constitute an important class among Riemannian manifolds, and whose study combines ideas from various areas of mathematics like geometry, topology, algebra, and mathematical analysis. In this thesis, we tackle the classification problem for homogeneous hypersurfaces in symmetric spaces. The results can be divided into two lines. The first of these consists in the development of a structural result for cohomogeneity one actions on symmetric spaces of noncompact type. This result guarantees that any such action can be constructed by one of five standard methods, easily described in terms of Lie algebras. The second line investigates cohomogeneity one actions on products of symmetric spaces of different types. Under certain hypotheses, one can reduce the study of these actions to each factor. This allowed us to produce a classification of codimension one homogeneous foliations on simply connected symmetric spaces.