A CONFORMAL DIFFERENTIAL INVARIANT AND THE CONFORMAL RIGIDITY OF HYPERSURFACES

For a hypersurface Vn-1 of a conformal space, we introduce a conformal differential invariant I = h(2)/g, where g and h are the first and th e second fundamental forms of Vn-1 connected by the apolarity conditio n. This invariant is called the conformal quadratic element of Vn-1. T he solution of the problem of conformal rigidity is presented in the f ramework of conformal differential geometry and connected with the con formal quadratic element of Vn-1. The main theorem states: Let n great er than or equal to 4, and let Vn-1 and <(V)over bar (n-1)> be two non isotropic hypersurfaces without umbilical points in a conformal space C-n or a pseudoconformal space C-q(n) of signature (p, q), p = n-q. Su ppose that there is a one-to-one correspondence f:Vn-1 --> <(V)over ba r (n-1)> between points of these hypersurfaces, and in the correspondi ng points of Vn-1 and <(V)over bar (n-1)> the following conditions hol ds: (I) over bar = fI, where f*:T(Vn-1) --> T(<(V)over bar (n-1)> is a mapping induced by the correspondence f. Then the hypersurfaces Vn-1 and <(V)over bar (n-1)> are conformally equivalent.