Einstein-like conditions and cosymplectic geometry

We prove that every Einstein compact almost C-manifold M^{2n+s} whose Reeb vector fields are Killing is a C-manifold. Then we extend this result considering some generalizations of the Einstein condition (η-Einstein, generalized quasi Einstein, etc.). Moreover, we find some topological properties of compact almost C-manifolds under the assumption that the Ricci tensor is transversally positive definite and the Reeb vector fields are Killing, namely we prove that the first Betti number is s and the first fundamental group is isomorphic to Z^{s}. Finally, a splitting theorem for cosymplectic manifolds is found.