Projective structures and contact forms

Abstract

In this paper we show that a projective structure on a real simply connected smooth manifold M determines a family of local Lie algebras on M (i.e., a family of Lie brackets on C ~ ( M ) that satisfy the localization condition). As was shown in Kirillov's paper [4], a nondegenerate local Lie algebra is determined by a contact or a symplectic structure on the manifold. The relation between projective and contact geometries has already been mentioned by Cartan (see [9] and also [2]). Many problems of projective differential geometry lead to contact structures. It turns out that this relationship is absolutely direct. Seemingly, the following elementary (and curious) fact remains unnoticed. T h e o r e m 1. Let M be a simply connected manifold (possibly noncompact) of dimension 2k 1. For a projective structure on M to exist it is necessary and sufficient that there be functions f l , . . . , f2k e C ~ ( M ) such that the 1-form k o~ = E ( f 2 i 1 d f 2 i f2idf2i-1) (1) i=1 is contact. Thus, a locally projective manifold of odd dimension is contact. The corresponding local Lie algebra is nondegenerate and is determined by a Lagrange bracket. The choice of the form (1) is not unique. An even-dimensional manifold with a projective structure must not be symplectic. However, this holds if the projective structure is affine. From this viewpoint, there is a sharp difference between even-dimensionai and odd-dimensional projective structures, while there is a strict analogy between "projective-contact" and "affine-symplectic" geometries. T h e o r e m 1'. For a simply connected manifold M of dimension 2k to possess an affine structure it is necessary and sufficient that there be functions f l , . . . , f2k E C¢~(M) such that the 2-form