Abstract. We give an up-to-date overview of geometric and topological properties of cosymplectic and coKähler manifolds. We also mention some of their applications to time-dependent mechanics.

We prove that on a compact Sasakian manifold (M, η, g) of dimension 2n+1, for any 0 ≤ p ≤ n the wedge product with η ∧(dη) p defines an isomorphism between the spaces of harmonic forms Ω n−p ∆ (M ) and Ω n+p+1 ∆ (M ). Therefore it induces an isomorphism between the de Rham cohomology spaces H n−p (M ) and H n+p+1 (M ). Such isomorphism is proven to be independent of the choice of a compatible Sasakian metric on a given contact manifold. As a consequence, an obstruction for a contact manifold to admit Sasakian structures is found.

We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian and 5-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact manifold. Focusing on the 5-dimensional case, we prove that there exists a one-to-one correspondence between nearly Sasakian structures and a special class of nearly hypo SU (2)-structures. By deforming such an SU (2)-structure, one obtains in fact a Sasaki-Einstein structure. Further we prove that both nearly Sasakian and Sasaki-Einstein 5-manifolds are endowed with supplementary nearly cosymplectic structures. We show that there is a one-to-one correspondence between nearly cosymplectic structures and a special class of hypo SU (2)-structures which is again strictly related to Sasaki-Einstein structures. Furthermore, we study the orientable hypersurfaces of a nearly Kähler 6-manifold, and in the last part of the paper, we define canonical connections for nearly Sasakian manifolds, which play a role similar to the Gray connection in the context of nearly Kähler geometry. In dimension 5, we determine a connection which parallelizes all the nearly Sasakian SU (2)-structure as well as the torsion tensor field. An analogous result holds also for Sasaki-Einstein structures.

Abstract. We prove that a compact nilmanifold admits a Sasakian structure if and only if it is a compact quotient of the generalized Heisenberg group of odd dimension by a co-compact discrete subgroup.

Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively