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.

Algebraic nature of singular Riemannian foliations in spheres

“…We refer the reader to [18], [3] and the literature therein for introductions to the subject. Note that the assumption that all leaves are compact makes further usual assumptions on M like compactness or completeness irrelevant.…”

Mean Curvature Flow of Singular Riemannian Foliations

“…[11, Theorem 6.1, Proposition 6.5], [2], [8]) that it is possible to find a neighbourhood P of q in L, a cylindrical neighbourhood O ǫ = T ub ǫ (P ) of q in M and diffeomorphism ϕ : O ǫ → V ⊆ T q M onto a neighbourhood V of the origin, such that:…”

The dual foliation of some singular Riemannian foliations

“…We recall some basic notions about singular Riemannian foliations, for further details we refer the readers to [1,8,13,16]. A singular Riemannian foliation F on a Riemannian manifold M is a decomposition of M into smooth injectively immersed submanifolds L(p), called leaves, such that it is a singular foliation and any geodesic starting orthogonally to a leaf remains orthogonal to all leaves it intersects.…”