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 measures how a deformation deviates from a one parameter family of motions up to 1st order. We implement this method to 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. This 1-rigidity is then extended to a local rigidity, which roughly states that if a CR submanifold $ M$ is Bochner rigid, then any CR submanifold that is sufficiently close and CR equivalent to $ M$ is congruent to $ M$ by an automorphism of the sphere.