On the Geometry of Almost -Manifolds

Abstract

An -structure on a manifold is an endomorphism field satisfying . We call an f-structure regular if the distribution is involutive and regular, in the sense of Palais. We show that when a regular f-structure on a compact manifold M is an almost -structure, it determines a torus fibration of M over a symplectic manifold. When rank , this result reduces to the Boothby-Wang theorem. Unlike similar results for manifolds with -structure or -structure, we do not assume that the f-structure is normal. We also show that given an almost -structure, we obtain an associated Jacobi structure, as well as a notion of symplectization.