An introduction to isoparametric foliations

Abstract

A hypersurface in a Riemannian manifold is called isoparametric if it and its nearby equidistant hypersurfaces have constant mean curvature. These geometric objects, as well as their important generalization to isoparametric submanifolds of codimension greater than one, appear in families called isoparametric foliations.

In these notes we present an introduction to the theory of isoparametric foliations in Riemannian geometry, starting from the problem in Geometric Optics that motivated their study, and then explaining the main results known so far, with focus on some recent techniques. We also include appendices on Jacobi field theory, isometric actions and symmetric spaces.

This text, which is based on a course taught in the framework of the Doctoral Program in Mathematics of the University of São Paulo in 2014, can serve as a reference for master and PhD students interested in Riemannian geometry.