Almost complex structures arising in contact geometry

Let E be a contact structure on a smooth manifold M defined by a global contact form alpha, and let J(alpha, (&phis;, g)) be an almost complex structure on M x R defined using a contact metric structure (&phis;, g) for alpha. We prove that the homotopy class [J] of J(alpha, (&phis;, g)) depends only on E , i.e. is independent of the choice of the contact metric structure (&phis;, g) and of alpha. We also prove that the space CME of all contact metric structures for all contact forms defining the contact structure E deforms to a single point, which is an invariant of the contact structure. We prove a generalization of the first result: Choose an embedding j : M → P into a symplectic manifold ( P, o) with j*o = dalpha. Any almost complex structure on P compatible with the symplectic form o induces on M a contact metric structure for a contact form, essentially equivalent to alpha, and determines an almost complex structure on M x R whose homotopy class depends only on E provided that Jj*alpha is a Liouville vector field, when xialpha is the Reeb field of xi alpha.