Differential Geometry and Analysis on CR Manifolds

? A CR manifold is aC differentiable manifold endowed with a complex subbundle T (M)ofthecomplexi?edtangentbundleT(M)?CsatisfyingT (M)?T (M)= 1,0 1,0 1,0 (0) and the Frobenius (formal) integrability property ? ? ? (T (M)), (T (M)) ? (T (M)). 1,0 1,0 1,0 ? The bundle T (M) is the CR structure of M, and C maps f : M ? N of CR 1,0 manifolds preserving the CR structures (i.e., f T (M)?T (N)) areCRmaps.CR ? 1,0 1,0 manifolds and CR maps form a category containing that of complex manifolds and holomorphic maps. The most interesting examples of CR manifolds appear, however, as real submanifolds of some complex manifold. For instance, any real hypersurface n M in C admits a CR structure, naturally induced by the complex structure of the ambient space 1,0 n T (M)=T (C)? [T(M)?C]. 1,0 1 n n Let(z,...,z) be the natural complex coordinates onC . Locally, in a neighborhood of each point of M, one may produce a frame{L :1?? ?n?1} ofT (M). G- ? 1,0 metrically speaking, eachL is a (complex) vector ?eld tangent to M. From the point ? of view of the theory of PDEs, the L ’s are purely tangential ?rst-order differential ? operators n