Progress in the Theory of Singular Riemannian Foliations

TL;DR

This survey provides a comprehensive overview of singular Riemannian foliations, covering foundational concepts, recent advances, and applications, including local structure, special classes, and desingularizations, with proof sketches and techniques.

Contribution

It offers an extensive introduction and synthesis of recent developments in SRF theory, including solutions to key problems and connections to other geometric structures.

Findings

Characterization of SRFs with orbifold leaf spaces

Solutions to the Palais-Terng problem on horizontal distribution

Applications to Molino's conjecture and desingularizations

Abstract

A singular foliation is called a singular Riemannian foliation (SRF) if every geodesic that is perpendicular to one leaf is perpendicular to every leaf it meets. A typical example is the partition of a complete Riemannian manifold into orbits of an isometric action. In this survey, we provide an introduction to the theory of SRFs, leading from the foundations to recent developments in research on this subject. Sketches of proofs are included and useful techniques are emphasized. We study the local structure of SRFs in general and under curvature conditions. We review the solution of the Palais-Terng problem on integrability of the horizontal distribution. Important special classes of SRFs, like polar and variationally complete foliations and their connections, are treated. A characterisation of SRFs whose leaf space is an orbifold is given. Moreover, desingularizations of SRFs are studied and applications, e.g., to Molino's conjecture, are presented.