A conformal differential invariant and the conformal rigidity of hypersurfaces

Abstract

For a hypersurface Vof a conformal space, we introduce a conformal differential invariant I = hg, where g and h are the first and the 9 second fundamental forms of V'1 connected by the apolarity condition. This invariant is called the conformal quadratic element of V'-1. The solution of the problem of conformal rigidity is presented in the framework of conformal differential geometry and connected with the conformal quadratic element of Vn-1. The main theorem states: Let n > 4, and let Vn- and Vn 1 be two nonisotropic hypersurfaces with- out umbilical points in a conformal space Cn or a pseudoconformal space C: of signature (p, q), p = n - q. Suppose that there is a one-to-one correspondence