Transverse LS category for Riemannian foliations

Abstract

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat e (M, F) is introduced in this paper, and we prove that cat e (M, F) is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category cat (M, F) is finite, and thus cat e (M, F) = cat(M, F) holds. A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of cat e (M, F) into a standard problem about O(q)-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated O(q)-manifold W, we have that cat e (M, F) = cat O(q) (Ŵ). Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. A generalization of the Lusternik-Schnirelmann theorem is derived: given a C 1 -function f: M → R which is constant along the leaves of a Riemannian foliation F, the essential transverse category cat e (M, F) is a lower bound for the number of critical leaf closures of f.