The equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics

We consider the class of Levi nondegenerate hypersurfaces M in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] that admit a local (CR transversal) embedding, near a point p ε M, into a standard nondegenerate hyperquadric in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] with codimension k := N - n small compared to the CR dimension n of M. We show that, for hypersurfaces in this class, there is a normal form (which is closely related to the embedding) such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric. We also show that if the signature of M and that of the standard hyperquadric in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] are the same, then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric.