Polar actions on symmetric spaces of noncompact type

Speaker: Juan Manuel Lorenzo-Naveiro 
Affiliation: CITMAGA - Universidade de Santiago de Compostela

Abstract

An isometric action of a Lie group on a complete Riemannian manifold is said to be polar if there exists a submanifold that intersects every orbit orthogonally. Such a submanifold is known as a section and is totally geodesic. These actions give a generalization of several well-known concepts in geometry, such as the polar coordinate system in the Euclidean plane or the spectral theorem for self-adjoint operators.

The aim of this talk is to give an overview of the classification problem for polar actions on symmetric spaces, focusing on those of noncompact type. Later, I will report on a joint work with J.C. Díaz-Ramos (Universidade de Santiago de Compostela) in which we classify polar homogeneous foliations on symmetric spaces of noncompact type whose section is homothetic to the hyperbolic plane. To this end, we will describe a method proposed by J. Berndt and H. Tamaru which allows us to extend submanifolds and isometric actions on hyperbolic spaces to arbitrary symmetric spaces, and show how the aforementioned foliations can be obtained from this procedure.