Transverse LS category for Riemannian foliations

Abstract:

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation $\mathcal {F}$ on a closed manifold $M$. The essential transverse category $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ is introduced in this paper, and we prove that $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category $\operatorname {cat}_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ is finite, and thus $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F}) = \operatorname {cat}_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ 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 $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ into a standard problem about $\mathbf O(q)$-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated $\mathbf O(q)$-manifold $\widehat W$, we have that $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F}) = \operatorname {cat}_{\mathbf O(q)}(\widehat W)$. 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 $f \colon M \to \mathbb {R}$ which is constant along the leaves of a Riemannian foliation $\mathcal {F}$, the essential transverse category $\operatorname {cat}^e_{\cap {\mkern -9mu}\mid }(M,\mathcal {F})$ is a lower bound for the number of critical leaf closures of $f$.