un 2 00 5 1 - rigidity of CR submanifolds in spheres

In the study of submanifolds in a homogeneous space X = G/P of a Lie group G, the method of moving frames is both a unifying concept and an effective tool, which is the version of the method of equivalence applied to submanifold geometry. Let φ be the left invariant, Lie algebra g-valued Maurer-Cartan form of G. The local equivalence problem for a submanifold f : M →֒ X is solved on a canonical adapted subbundle Ef : Bf →֒ G together with π = E f φ in such a way that the complete set of local invariants is generated by the coefficients of π [Ga]. The pair (Bf , π) measures in a sense how the submanifold M deviates from a flat model submanifold. The method was systematically exploited and applied by Cartan himself and by Chern in various geometric problems, [Ch] and the references therein.