TRANSVERSE LS CATEGORY FOR RIEMANNIAN FOLIATIONS

W study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat (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) over cap, we have that cat(e), (M,F) = cat(O(q))((W) over cap) 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