International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 3, Pages 521-536

Locally conformal symplectic manifolds

Izu Vaisman

Department of Mathematics, University of Haifa, Israel

Received 4 April 1984

Copyright © 1985 Izu Vaisman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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, dΩα=0. Equivalently, dΩ=ωΩ 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=dθωθ. 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.