On locally conformal symplectic manifolds of the first kind

Abstract

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

On a certain class of locally conformal symplectic structures of the second kind

“…In [2] the authors classify the 4-dimensional solvable Lie algebras admitting LCS structures up to automorphism of the Lie algebra. Using their classification we show in Table 2 all the 4-dimensional solvable Lie algebras with their associated LCS structure satisfying condition (4), which means that they can be extended to a higher dimensional unimodular Lie algebra admitting a LCS structure given by Theorem 3.3.…”

Locally conformally Kähler solvmanifolds: a survey

“…Recently, in [23] and [10] many interesting results on LCS structures of the first kind on Lie algebras and solvmanifolds were established. We review some of them next.…”

“…Theorem 7.2 ( [23]). There is a one to one correspondence between Lie algebras of dimension 2n + 2 admitting a LCS structure of the first kind with central Lee vector and symplectic Lie algebras (s, β) of dimension 2n endowed with a derivation E such that β(EX, Y )+β(X, EY ) = 0 for all X, Y ∈ s.…”

“…Considering LCS structures on nilpotent Lie algebras, it was shown in [23] that any such structure is of the first kind and, moreover, the Lee vector is central. Combining this with the previous theorem the next result follows: Moreover, the symplectic nilpotent Lie algebra s may obtained by a sequence of (n − 1) symplectic double extensions by nilpotent derivations from the abelian Lie algebra of dimension 2, according to [66,34].…”

On the topology of metric f–K-contact manifolds

“…For instance, the mapping torus of a symplectomorphism of a symplectic manifold naturally is a cosymplectic manifold, and that of a holomorphic isometry of a Kähler manifold is a co-Kähler manifold (see[21, Lemmata 1 and 4]). The mapping torus of a strict contactomorphism of a contact manifold is a locally conformally symplectic manifold[4, Example 2.4], and an automorphism of a Sasakian manifold induces a Vaisman structure on the mapping torus. From this point of view, metric f -K-contact structures behave in a rather unusual way.…”