K-Cosymplectic manifolds

• Published in: Annals of global analysis and geometry Vol. 47; no. 3; pp. 239 - 270
• Main Authors: Bazzoni, Giovanni, Goertsches, Oliver
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2015; Springer Nature B.V
• Subjects: Analogies; Analysis; Bundling; Contact; Deformation; Differential Geometry; Fields (mathematics); Flats; Geometry; Global Analysis and Analysis on Manifolds; Killing; Manifolds; Mathematical Physics; Mathematics; Mathematics and Statistics; Orbits; Deformations; 55N91; Closed Reeb orbit; Secondary 53C12; 53D20; Cosymplectic; 53D15; Basic cohomology; Momentum map; Primary 53C25
• ISSN: 0232-704X; 1572-9060
• DOI: 10.1007/s10455-014-9444-y

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 coKähler 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 Kähler 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.