On locally conformal symplectic manifolds of the first kind

TL;DR

This paper explores examples and structure theorems of locally conformal symplectic manifolds of the first kind, focusing on nilmanifolds and Lie groups, revealing their geometric and topological properties.

Contribution

It provides new examples of such structures, especially on nilmanifolds, and offers a comprehensive classification of left-invariant structures on nilpotent Lie groups.

Findings

Examples of locally conformal symplectic structures on nilmanifolds that lack Vaisman metrics

A structure theorem characterizing these manifolds under topological restrictions

Complete description of such structures on nilpotent Lie groups via extensions

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 K\"ahler 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.