1-rigidity of CR submanifolds in spheres

We propose a unified computational framework for the problem of deformation and rigidity of submanifolds in a homogeneous space under geometric constraint. A notion of 1-rigidity of a submanifold under admissible deformations is introduced. It means every admissible deformation of the submanifold osculates a one parameter family of motions up to 1st order. We implement this idea to the question of rigidity of CR submanifolds in spheres. A class of submanifolds called Bochner rigid submanifolds are shown to be 1-rigid under type preserving CR deformations. 1-rigidity is then extended to a rigid neighborhood theorem, which roughly states that if a CR submanifold M is Bochner rigid, then any pair of mutually CR equivalent CR submanifolds that are sufficiently close to M are congruent by an automorphism of the sphere. A local characterization of Whitney submanifold is obtained, which is an example of a CR submanifold that is not 1-rigid. As a by product, we give a simple characterization of the proper holomorphic maps from the unit ball B to B.