LOCALLY CONFORMAL SYMPLECTIC MANIFOLDS

. A locally conformal symplectic (l.c.s.) manifold is a pair (M2n,fl) where M2n(n > i) is a connected differentiable manifold, and a nondegenerate 2-form on M such that M

interesting manifolds. In between, the locally conformal symplectlc (l.c.s.) manifolds are defined as almost symplectic manifolds M 2n (n > I) which  O. Clearly, iff is exact, the manifold is globally conformal symplectic (g.c.s.).
Therefore, the l.c.s, manifolds are natural phase spaces of Hamiltonlan dynamical systems, more general then the symplectic manifolds, and this is the announced motivation.
Finally, we indicate that the l.c.s, manifolds play an important role in the recent works of A. Lichnerowicz . ].
In this paper, we do not intend to discuss problems of mechanics or physics, but some problems concerning the differential geometrical structure of the l.c.s, manifolds. In Bectlon     ),and since exptB) preserves it also preserves A. This means [B,A] O, and we obtain on M the vertical foliation V span {A,B} whose leaves are the orbits of a natural action of 2 In the next Section, we shall use V in order to get more geometric information on M.
In connection with the above discussion, we shall also make the following complementary considerations. Formula (2.6) proves that a Hamiltonian field is a conformal infinitesimal transformation(c.i.t.)of (M,). Generally, a vector field X of M and we shall say that -M has many c.i.t. Hence, by comparing with (2.9), we see that an l.c.s, manifold with many c.i.t, is a candidate of a manifold of the first kind. More precisely, let us note that the Hence the existence of many c.l.t, is a conformally invariant property. Now, assume that there is a vector field C on (M,) such that (C)   (8)  It is easy to deduce from this that the tangent distribution of the leaves of on G/K is precisely the vertical distribution of the fibration p But then, by applying the Corollary on p. 28 of [14] it follows that the foliation V is simple and its basis is a covering manifold of G/H i.e., a Hausdorff manifold.

REMARKS. i) Just like in 8
we can see that the space of the leaves

COMPACT HOMOGENEOUS L.C.S. MANIFOLDS.
Like for the regularity property, we may expect to obtain information about cornpact homogeneous l.c.s, manifolds from a discussion of compact honogeneous s-contact manifolds, and for the latter it is possible to extend in a rather straightforward manner the results established for contact manifolds in [I0 ]. Obviously, N is also compact and symplectic homogeneous with the homogeneity group G since G preserves the whole structure of M and, particularly, the vertical foliation V (see Section 3 for notation). Now, recall that a homogeneous symplectic manifold with group G is homogeneous strongly symplectic if for every _X 6 g the field X N induced on N is a Hamiltonian field. Then we have  Here, only the last assertion has to be justified, and it follows by first considering M as a flat circle bundle over a contact manifold P, then fiberinK P over N as a principal fibre bundle, and finally by applying twice the Gysin exact sequence theorem and using the fact that bl(N) 0 Now, let us consider again a homogeneous l.c.s, manifold (M G/K, ), let g hot be the Lie algebra of G and g the subalgebra of those X of g that m()--O. and q g(p), g G,g expXlO...exp h' XI,... E g ,then, generally, the situation is such that, for instance, XI ghor, X2 ghor, X3 hor etc. But then exp X I sends the leaf L to a leaf L', exp X_2 sends L' to L" exp X 3 preserves L" etc., and we also must have some X_u, X_v, X such that their exponentials bring us back from L" (or whatever other leaf) to L.
Since any bracket of i.a. is horizontal, if we exchange in g the order of the exponentials such that X ,X_v,X_w come next to X 1 X 2 this adds a factor in G hr Then, exp XlOexp X2=ex p X =exD X =exp X preserves L and is also in G hr There- Let M m be a compact connected Riemannian manifold with the metric g Let us assume that there is given an action of the additive group IR s on M m by isometries of g all of whose orbits are s-dimensional. Then, the orbits of this action define on M a foliation V (called the vertical foliation) whose leaves are s-dimensional submanifolds tangent to some independent commuting vector fields E (u i,..., s) u provided by the natural basis of ]R s Clearly, we have u g 0 If V is a simple foliation whose space of leaves is Hausdorff, we say that the action of IR s on M is regular.
A few more simple details about (M,g) and the action above will be needed. Namely, let C be the horizontal distribution orthogonal to Then, we can define the 2-tensor and the s 1-forms y(X,Y) g(pr C X, pr C Y) PROOF. The proof of the existence of the principal bundle structure required is exactly the same as in the case of Proposition 3.1. All the other facts stated in Proposition are easy consequences of the formulas (i) (6).