Locally conformal symplectic manifolds

Abstract

A locally conformal symplectic (l. c. s.) manifold is a pair (M2n,Ω) where M2n(n>1) is a connected differentiable manifold, and Ω a nondegenerate 2-form on M such that M=⋃αUα (Uα- open subsets). Ω/Uα=eσαΩα, σα:Uα→ℝ, dΩα=0. Equivalently, dΩ=ω∧Ω for some closed 1-form ω. L. c. s. manifolds can be seen as generalized phase spaces of Hamiltonian dynamical systems since the form of the Hamilton equations is, in fact, preserved by homothetic canonical transformations. The paper discusses first Hamiltonian vector fields, and infinitesimal automorphisms (i. a.) on l. c. s. manifolds. If (M,Ω) has an i. a. X such that ω(X)≠0, we say that M is of the first kind and Ω assumes the particular form Ω=dθ−ω∧θ. Such an M is a 2-contact manifold with the structure forms (ω,θ), and it has a vertical 2-dimensional foliation V. If V is regular, we can give a fibration theorem which shows that M is a T2-principal bundle over a symplectic manifold. Particularly, V is regular for some homogeneous l. c. s, manifolds, and this leads to a general construction of compact homogeneous l. c. s, manifolds. Various related geometric results, including reductivity theorems for Lie algebras of i. a. are also given. Most of the proofs are adaptations of corresponding proofs in symplectic and contact geometry. The paper ends with an Appendix which states an analogous fibration theorem in Riemannian geometry.

Keywords

• locally conformal symplectic manifold
• s-contact manifold
• Boothby-Wang fibration.