Leaf closures of Riemannian foliations: a survey on topological and geometric aspects of Killing foliations

TL;DR

This survey explores the topological and geometric properties of Killing foliations, focusing on leaf closures, Molino's structural theory, and recent advances, providing a comprehensive overview for a broad mathematical audience.

Contribution

It offers a detailed introduction to Killing foliations, connecting classical and recent results, and discusses singular cases, enhancing understanding of their structure and properties.

Findings

Description of leaf closures via transverse Killing flows

Connection between Molino's theory and pseudogroup structures

Overview of recent developments in singular Killing foliations

Abstract

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino's structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations and discuss singular Killing foliations.