An almost cosymplectic manifold M is a (2m+1)-dimensional oriented Riemannian
manifold endowed with a 2-form Ω of rank 2m, a 1-form η such that Ωm Λ η≠0 and a vector field ξ
satisfying iξΩ=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.
