Transverse LS-Category for Riemannian Foliations

TL;DR

This paper investigates the transverse Lusternik-Schnirelmann category in Riemannian foliations, providing conditions for finiteness, introducing an essential variant, and relating it to critical leaf closures.

Contribution

It introduces the essential transverse category, proves its finiteness for all Riemannian foliations, and establishes its relation to critical leaf closures.

Findings

Finite transverse LS category characterized by specific conditions.

Essential transverse category is finite for all Riemannian foliations.

Lower bound for critical leaf closures given by the essential transverse category.

Abstract

We study the transverse Lusternik-Schnirelmann category of a Riemannian foliation on a compact manifold. We obtain a necessary and sufficient condition when the transverse LS category is finite. We also introduce a variation on the concept of transverse LS category, the essential transverse category, and show that this is finite for every Riemannian foliation and coincides with the transverse category if the latter is finite. Moreover we prove that the essential transverse category is a lower bound for the number of critical leaf closures of a basic differentiable function on M.