K-cosymplectic manifolds

TL;DR

This paper explores the geometry and topology of K-cosymplectic manifolds, generalizing coKähler structures, analyzing their cohomology, vector fields, group actions, and deformations, with implications for regularity and Reeb orbit counts.

Contribution

It introduces the concept of K-cosymplectic manifolds, studies their cohomology, vector fields, and group actions, and establishes results on their structure and deformations.

Findings

Regular K-cosymplectic manifolds are flat circle bundles over almost K"ahler manifolds.

Deformations of type I show all compact K-cosymplectic manifolds have quasi-regular structures.

Relations between basic cohomology and Reeb orbits are established for irregular cases.

Abstract

In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coK\"ahler structures, in the same way as K-contact structures generalize Sasakian structures. In analogy to the contact case, we distinguish between (quasi-)regular and irregular structures; in the regular case, the K-cosymplectic manifold turns out to be a flat circle bundle over an almost K\"ahler manifold. We investigate de Rham and basic cohomology of K-cosymplectic manifolds, as well as cosymplectic and Hamiltonian vector fields and group actions on such manifolds. The deformations of type I and II in the contact setting have natural analogues for cosymplectic manifolds; those of type I can be used to show that compact K-cosymplectic manifolds always carry quasi-regular structures. We consider Hamiltonian group actions and use the momentum map to study the equivariant cohomology of the canonical torus action on a compact K-cosymplectic manifold, resulting in relations between the basic cohomology of the characteristic foliation and the number of closed Reeb orbits on an irregular K-cosymplectic manifold.