1-rigidity of CR submanifolds in spheres

Details 1-rigidity of CR submanifolds in spheres 1-rigidity of CR submanifolds in spheres

2005-06-08

arxiv.org/abs/math/0506134v2

Mathematics

Differential Geometry

43 pages

Scientific paper

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.

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