On Three-dimensional Cauchy-riemann Manifolds

of the almost complex tensor of C2 ("multiplication by i "). The collection of these subspaces HpM forms a bundle H MeT M called the horizontal or contact bundle. (Indeed, strict pseudoconvexity of M implies that the plane field {HpM} is nondegenerate, i.e. defines a contact structure.) The almost complex tensor of C2 then restricts to a bundle endomorphism J : H M -+ H M such that J2 = -id. The bundle H MeT M together with this endomorphism J defines the CR structure of M. Slightly more generally, let D = DuM be a four dimensional compact manifold with boundary M, interior D, endowed with a smooth almost complex tensor which is integrable on D. If D is strictly pseudoconvex-i.e., in a neighborhood of any p E M there is a smooth strictly plurisubharmonic function U, negative on D, 0 on M, but du i= 0 on M-then M inherits a strictly pseudoconvex CR structure from D. In this case we shall say that M bounds a strictly pseudoconvex surface. This leads us to the abstract definition. A strictly pseudoconvex (threedimensional) CR manifold is a compact manifold M (without boundary), dim M = 3 , endowed with a contact structure H M = {HpM : p E M} c T M and an endomorphism J of H M such that J2 = id . (Throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects. Thus M, H M , J are assumed to be smooth in this sense.) Furthermore, a differentiable function f: M -+ C is a CR function if for any p E M the restriction