A Generalization of the Boothby-wang Theorem

Abstract

We consider a Riemannian manifold $M$ with an $f$-structure. With some additional properties such a manifold is called a $\mathscr{K}, \mathscr{C} or \mathscr{S}$-manifold. The considered structures determine a Riemannian foliation, whose leaf closures form a singular Riemannian foliation. We give conditions under which the foliation of the principal stratum is again associated to a structure of the type we consider. The manifold can be partitioned into strata on which the leaf closures are given by toroidal fiber bundles. This theorem is a topological generalization of the classical Boothby-Wang theorem for the contact manifolds.