A RELATIVE INDEX ON THE SPACE OF EMBEDDABLE CR-STRUCTURES, I

We study the. problem of embeddability for three dimensional CR-manifo lds. Let (M, (0) partial derivative(b)) denote a compact, embeddable, strictly pseudoconvex CR-manifold and S-0 the orthogonal projection on ker (0) partial derivative(b). If (1) partial derivative(b) denotes a deformation of this CR-structure then (1) partial derivative(b) is em beddable if and only if S-0: ker(1) partial derivative(b) --> ker(0) p artial derivative(b) is a Fredholm operator. We define the relative in dex, Ind((0) partial derivative(b), (1) partial derivative(b)), to be the Fredholm index of this operator. This integer is shown to be indep endent of the volume form used to define S-0 and to be constant along orbits of the group of contact transformations. The relative index the refore defines a stratification of the moduli space of embeddable CR-s tructures. For small perturbations its value is related to small eigen values of the associated square(b)-operator.