arXiv subjects
Publications and source records attributed to Carlos Olmos.
In this paper we give a geometric proof of the Karpelevich's theorem that asserts that a semisimple Lie subgroup of isometries, of a symmetric space of non compact type, has a totally geodesic orbit. In fact, this is equivalent to a well-known result of Mostow about existence of compatible Cartan decompositions.
We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In particular, we develop some of the structural theory of copolarity k representations, we classify the irreducible representations of copolarity one, and we relate the copolarity of an isometric action to the concept of variational completeness in the sense of Bott and Samelson.
We define and study isoparametric submanifolds of general ambient spaces and of arbitrary codimension. In particular we study their behaviour with respect to Riemannian submersions and their lift into a Hilbert space. These results are used to prove a Chevalley type restriction theorem which relates by restriction eigenfunctions of the Laplacian on a compact Riemannian manifold which contains an isoparametric submanifold with flat sections to eigenfunctions of the Laplacian of a section. A simple example of such an isoparametric foliation is given by the conjugacy classes of a compact Lie group and in that case the restriction theorem is a (well known) fundamental result in representation theory. As an application of the restriction theorem we show that isoparametric submanifolds with flat sections in compact symmetric spaces are level sets of eigenfunctions of the Laplacian and are hence related to representation theory. In addition we also get the following results. Isoparametric submanifolds in Hilbert space have globally flat normal bundle, and a general result about Riemannian submersions which says that focal distances do not change if a submanifold of the base is lifted to the total space.