On closed geodesics in the leaf spaces of singular Riemannian foliations

TL;DR

This paper reviews recent advances in the study of singular Riemannian foliations and orbifolds, demonstrating the existence of closed geodesics in their leaf spaces under specific conditions and exploring the shortening process.

Contribution

It combines recent results to establish the existence of closed geodesics in leaf spaces of certain singular Riemannian foliations, a novel synthesis in the field.

Findings

Existence of closed geodesics in leaf spaces with sections

Existence of closed geodesics in leaf spaces without horizontal conjugate points

Analysis of the shortening process in Riemannian foliations

Abstract

In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points. We also investigate the shortening process with respect to Riemannian foliations.