Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms

Abstract

LetM andN be real analytic (C) CR manifolds of hypersurface type of dimensions 2m+1 and 2n+1, m ≤ n, respectively. In this paper we show under generic assumptions that a CR mapping f : M → N satisfies a certain Pfaffian system in the jet space, which implies the analyticity and rigidity of CR mappings. We use a method of prolongation for the tangential Cauchy-Riemann equations. The technique is, roughly speaking, separating the holomorphic derivatives of CR functions from their complex conjugates and applying the tangential Cauchy-Riemann operators and then counting the order of the missing-directional derivatives in the holomorphic part. The argument of this paper is purely local, thus for instance, a manifold should be understood as a germ of a manifold at a reference point and mappings are supposed to preserve the reference point. First, we recall some basic definitions. For other definitions and proofs of the facts that we do not present here the readers are refered to [Jac]. Let M be a differentiable manifold of dimension 2m+1. A CR structure on M is a subbundle V of the complexified tangent bundle TCM having the following properties : i) each fiber is of complex dimension m, ii) V ∩ V = { 0 }, iii) [ V,V ] ⊂ V (integrability). Given a CR structure V we have Levi form