We present some examples of locally conformal symplectic structures of the first kind on compact nilmanifolds which do not admit Vaisman metrics. One of these examples does not admit locally conformal Kähler metrics and all the structures come from left-invariant locally conformal symplectic structures on the corresponding nilpotent Lie groups. Under certain topological restrictions related with the compactness of the canonical foliation, we prove a structure theorem for locally conformal symplectic manifolds of the first kind. In the non compact case, we show that they are the product of a real line with a compact contact manifold and, in the compact case, we obtain that they are mapping tori of compact contact manifolds by strict contactomorphisms. Motivated by the aforementioned examples, we also study left-invariant locally conformal symplectic structures on Lie groups. In particular, we obtain a complete description of these structures (with non-zero Lee 1-form) on connected simply connected nilpotent Lie groups in terms of locally conformal symplectic extensions and symplectic double extensions of symplectic nilpotent Lie groups. In order to obtain this description, we study locally conformal symplectic structures of the first kind on Lie algebras. MSC classification [2010]: 22E25, 22E60, 53C12, 53D05, 53D10, 53C55

Abstract. We give a classification of minimal algebras generated in degree 1, defined over any field k of characteristic different from 2, up to dimension 6. This recovers the classification of nilpotent Lie algebras over k up to dimension 6. In the case of a field k of characteristic zero, we obtain the classification of nilmanifolds of dimension less than or equal to 6, up to k-homotopy type. Finally, we determine which rational homotopy types of such nilmanifolds carry a symplectic structure.

By the work of Li, a compact co-Kähler manifold M is a mapping torus K ϕ , where K is a Kähler manifold and ϕ is a Hermitian isometry. We show here that there is always a finite cyclic cover M of the form M ∼ = K × S 1 , where ∼ = is equivariant diffeomorphism with respect to an action of S 1 on M and the action of S 1 on K × S 1 by translation on the second factor. Furthermore, the covering transformations act diagonally on S 1 , K and are translations on the S 1 factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle. MSC classification [2010]: Primary 53C25; Secondary 53B35, 53C55, 53D05.Key words and phrases: co-Kähler manifolds, mapping tori.Let (M 2n+1 , J, ξ, η, g) be an almost contact metric manifold given by the conditionsThe authors of [CDM] use the term cosymplectic for Li's co-Kähler because they view these manifolds as odd-dimensional versions of symplectic manifoldseven as far as being a convenient setting for time-dependent mechanics [DT]. Li's characterization, however, makes clear the true underlying Kähler structure, so we have chosen to follow his terminology.