On the topology of metric f\u2013K-contact manifolds

Oliver Goertsches,Eugenia Loiudice

doi:10.1007/s00605-020-01400-z

Abstract

We observe that the class of metric f–K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f–K-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.

Highlights

An f -structure on a smooth manifold is a (1, 1)-tensor f of constant rank, satisfying f 3 + f = 0. This notion was introduced by Yano in [25] and generalizes both the notion of almost complex and of almost contact structure
The rank of f is always even, and if maximal, f is either an almost complex or an almost contact structure. f -structures with non-maximal rank (in particular with dim ker( f ) = 2) arise naturally when studying hypersurfaces of almost contact manifolds
An analogue of Hermitian structures on almost complex manifolds and of contact metric structures on almost contact manifolds was introduced on the class of Communicated by Andreas Cap

Summary

Introduction

We prove a splitting theorem for the de Rham cohomology of metric f –K contact manifolds, i.e., metric f -contact manifold M whose characteristic vector fields ξ1, . Theorem 1.1 For any compact metric f –K -contact manifold M there is an isomorphism of (Rs−1)-algebras. Ξs on a metric f –K -contact manifold M to the basic cohomology H ∗(M, F). Theorem 1.3 The characteristic foliation of a compact metric f –K -contact manifold M2n+s has at least n + 1 closed leaves. If it has only finitely many closed leaves, the following conditions are equivalent:. As a consequence we obtain that any automorphism of the K -contact structure on a K -manifold M2n+1 which has exactly n + 1 closed orbits sends every closed Reeb orbit to itself (see Corollary 6.5)

Metric f -manifolds

Mapping tori of metric f –K -contact manifolds

Cohomology of metric f –K -contact manifolds

Morse theory on metric f –K -contact manifolds

Closed leaves of the characteristic foliation