On a class of exact locally conformal cosymlectic manifolds

An almost cosymplectic manifold M is a ( 2 m + 1 ) -dimensional oriented Riemannian manifold endowed with a 2-form Ω of rank 2 m , a 1-form η such that Ω m Λ η ≠ 0 and a vector field ξ satisfying i ξ Ω = 0 and η ( ξ ) = 1 . Particular cases were considered in [3] and [6]. Let ( M , g ) be an odd dimensional oriented Riemannian manifold carrying a globally defined vector field T such that the Riemannian connection is parallel with respect to T . It is shown that in this case M is a hyperbolic space form endowed with an exact locally conformal cosymplectic structure. Moreover T defines an infinitesimal homothety of the connection forms and a relative infinitesimal conformal transformation of the curvature forms. The existence of a structure conformal vector field C on M is proved and their properties are investigated. In the last section, we study the geometry of the tangent bundle of an exact locally conformal cosymplectic manifold.