International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 3, Pages 521-536
Locally conformal symplectic manifolds
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 where
is a connected differentiable manifold, and a nondegenerate -form on such that (- open subsets). , , . Equivalently, for some closed -form . 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 has an i. a. such that , we say that is of the first kind and assumes the particular form . Such an is a -contact manifold with the structure forms , and it has a vertical -dimensional foliation . If is regular, we can give a fibration theorem which shows that is a -principal bundle over a symplectic manifold. Particularly, 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.