Transverse Lusternik–Schnirelmann category of Riemannian foliations

The transverse Lusternik–Schnirelmann category of a foliation is an invariant of foliated homotopy type. In this paper we show that the category of a Riemannian foliation is infinite if there exists a non-compact leaf verifying certain conditions. Examples of the obstruction to construct categorical coverings in such foliations are given.