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.

50 pages, no figures. Contains and completes the note arXiv:1407.5510. New results on locally conformal symplectic Lie groups. Comments very welcome!

References in corpus (4)

• LCK rank of locally conformally Kahler manifolds with potential
• Moser stability for locally conformally symplectic structures
• Vaisman nilmanifolds
• Universal Models via Embedding and Reduction for Locally Conformal Symplectic Structures

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• Almost formality of quasi-Sasakian and Vaisman manifolds with applications to nilmanifolds
• Hard Lefschetz Theorem for Vaisman manifolds
• On certain class of locally conformal symplectic structures of the second kind
• Locally conformal symplectic structures on Lie algebras of type I and their solvmanifolds
• Special types of locally conformal closed G${}_2$-structures
• The convexity package for Hamiltonian actions on conformal symplectic manifolds