Progress in the Theory of Singular Riemannian Foliations

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.

References in corpus (5)

• Proofs of Conjectures about singular riemannian foliations
• Desingularization of singular Riemannian foliation
• Singular riemannian foliations with sections, transnormal maps and basic forms
• On polar foliations and fundamental group
• Polar foliations and isoparametric maps

Cited by in corpus (9)

• Singular Riemannian foliations and applications to positive and nonnegative curvature
• The Basic de Rham Complex of a Singular Foliation
• On singular Finsler foliation
• Geodesics on Riemannian stacks
• Isospectral Alexandrov Spaces
• A-Foliations of codimension two on compact simply-connected manifolds
• Core reduction for singular Riemannian foliations in positive curvature
• Singular Riemannian flows and characteristic numbers
• A Weyl's Law for Singular Riemannian Foliations with Applications to Invariant Theory