Some Global Properties of Contact Structures

Abstract

transformations in general and to the study of global contact transformations in the special case of euclidean space. In attempting to generalize Lie's results to more general manifolds, it becomes clear that there are intrinsic global differences between the even and odd dimensional cases. In this paper, only the odd dimensional case will be discussed. Intuitively, a manifold carries a contact structure if the coordinate transformations can be chosen to preserve the 1-form dz - y'dx' up to a non-zero, multiplicative factor. We first show that if the manifold is orientable then this property is equivalent to the existence of a globally defined 1form ca of maximal rank, and then we derive some further equivalent conditions. It is well known that the existence of such a 1-form implies that the structure group of the tangent bundle can be reduced to the unitary group. (See, e.g., Chern, [7]). If this can be done, we say that the manifold is an almost-contact manifold. The obstructions to the existence of such a structure are investigated and it is shown that the primary obstruction is the third Stiefel-Whitney class. This solves completely the question of the existence of U(2) structures on five dimensional manifolds. We turn then to a discussion of global contact transformations, i.e., transformations which preserve ca up to a non-zero, multiplicative factor, T. Lie's results are shown to be valid in general by providing intrinsic proofs of his theorems. It should be noted that in this context, in general, analysis occurs only in the definitions, while the proofs consist simply of algebraic manipulations. Sheaves are employed at this point only because they provide a convenient language. Finally, we show that the factors T which can occur in contact transformations are not arbitrary. These results are then applied to the study of deformations (in the