Locally conformal symplectic manifolds

Izu Vaisman

doi:10.1155/s0161171285000564

Abstract

A locally conformal symplectic (l. c. s.) manifold is a pair(M2n,Ω)whereM2n(n>1)is a connected differentiable manifold, andΩa nondegenerate2-form onMsuch thatM=⋃αUα(Uα- open subsets).Ω/Uα=eσαΩα,σα:Uα→ℝ,dΩα=0. Equivalently,dΩ=ω∧Ωfor some closed1-formω. 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.Xsuch thatω(X)≠0, we say thatMis of the first kind andΩassumes the particular formΩ=dθ−ω∧θ. Such anMis a2-contact manifold with the structure forms(ω,θ), and it has a vertical2-dimensional foliationV. IfVis regular, we can give a fibration theorem which shows thatMis aT2-principal bundle over a symplectic manifold. Particularly,Vis 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.

Highlights

> i) is a connected differentiable manifold, and a nondegenerate 2-form on o M such that M k9 U s (Us- open subsets) /U e
L.c.s. manifolds can be seen as generalized phase spaces of Hamiltonian dynamical systems since the form of the Hamilton equations is, preserved by homothetlc canonical transformations
X such that (X) 0, we say that M is of the first kind and assumes the particular form de ^ e

Summary

A’ and a system of functions

Dos glue up to a closed 1-form and such that d(e ) =0. Formula (i.i) was established by H.C. Its phase space can be seen as a 2n-dimensional differentiable manifold M and the dynamics consists of the orbits of a well defined vector field X. H(a) ()) and momenta, and there is a Hamiltonian function (qi (a), Pj given by positions such that the orbits are defined by the Hamilton equations i dq(e) H(s (c) dPi. the usual continuation of this interpretation consists in asking the local (s) H(e) forms and local functions to glue up to a global symplectic form and a global Hamiltonian H. The usual continuation of this interpretation consists in asking the local (s) H(e) forms and local functions to glue up to a global symplectic form and a global Hamiltonian H This is not compulsory since the only global entity is.

A and a corresponding system of local symplectic forms

The homogeneity of this structure will be proven like for the contact case in

E ghot such that

RIEMANNIAN MANIFOLDS