Homogeneous submanifolds and isoparametric hypersurfaces in symmetric spaces of non-compact type

Symmetry lies at the very core of science. Indeed, geometry was defined by Felix Klein as the study of those properties in a space that are invariant under a given transformation symmetry group.

In Riemannian geometry, this group is the isometry group, that is, the group of distance- preserving transformations of a given Riemannian manifold.  The action of a subgroup of the isometry group of a given manifold is called an isometric action.  Its cohomogeneity is the lowest codimension of the orbits.   An orbit  whose codimension  is greater than the cohomogeneity of the action is called a singular orbit.  The orbits of maximal dimension are called regular.  A submanifold is said to be (extrinsically) homogeneous if it is an orbit of the action of a subgroup of the isometry group on the ambient manifold.

In this thesis, we study certain kind of geometrical objects from the viewpoint  of their symmetries in the context of symmetric spaces of non-compact type.

The problem of classifying homogeneous hypersurfaces  (equivalently cohomogeneity one actions up to orbit equivalence) in Euclidean spaces stems from Geometrical Optics and traces back to the work by Somigliana at the beginning of the 20th century. Incidentally, his result initiated  the study of one of the geometric objects of interest in this thesis: isoparametric hypersurfaces.

Indeed, one of the main contributions of this thesis is the classification  of isoparametric hypersurfaces in complex hyperbolic spaces. In order to prove such classification,  we have also dealt with these geometrical objects in the anti-De Sitter space. Moreover, in this space, we have classified spacelike isoparametric hypersurfaces.

Cohomogeneity one actions have been usually investigated from a Lie-theoretic point  of view or from the viewpoint of their regular orbits (homogeneous hypersurfaces) and related concepts (such as isoparametric hypersurfaces). However, it is also interesting to approach cohomogeneity one actions with  regard to their singular orbits.  In fact, if one considers a cohomogeneity one action with a singular orbit on a connected complete Riemannian manifold, then the principal curvatures of this singular orbit, counted with multiplicities, do not depend on the normal directions.  In this thesis, a submanifold is called CPC if it has this geometric property of singular orbits of cohomogeneity one actions.  Although this condition on the principal curvatures is remarkably simple and natural, there seems to be no systematic study of CPC submanifolds in a general setting.

Therefore, in this thesis we present a systematic approach to the construction, description and classification of homogeneous CPC submanifolds in irreducible  Riemannian symmetric spaces of non-compact type and rank greater or equal than 2.  In order to do so, we have developed an original technique that allows us to calculate the Levi-Civita  connection in a very efficient way. This technique is based on the examination of the restricted root system of a symmetric space of non-compact type by means of the generalization of the concept of string. We expect this tool to be used or adapted to address different problems.

In particular, we have utilized it to study a class of intermediate submanifolds between CPC submanifolds and minimal submanifolds, that is, the class of austere submanifolds. A submanifold is said to be austere  if, at every point, the principal curvatures (counted with multiplicities)  with respect to any unit normal vector are invariant under change of sign. In the final part of this thesis, we have classified certain austere orbits arising from the theory of parabolic subgroups by a case-by-case analysis  of the root system of the symmetric space of non-compact type under consideration.