ar X iv : m at h / 98 06 00 1 v 1 [ m at h . D G ] 3 0 M ay 1 99 8 A CONFORMAL DIFFERENTIAL INVARIANT AND THE CONFORMAL RIGIDITY OF HYPERSURFACES

For a hypersurface V n−1 of a conformal space, we introduce a conformal differential invariant I = h 2 g , where g and h are the first and the second fundamental forms of V n−1 connected by the apolarity condition. This invariant is called the conformal quadratic element of V n−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 V n−1. The main theorem states: Let n ≥ 4 and V n−1 and V n−1 be two nonisotropic hypersurfaces without umbilical points in a conformal space C n or a pseudoconformal space C n q of signature (p, q), p = n − q. Suppose that there is a one-to-one correspondence f : V n−1 → V n−1 between points of these hypersurfaces, and in the corresponding points of V n−1 and V n−1 the following condition holds: I = f * I, where f * : T (V n−1) → T (V n−1) is a mapping induced by the correspondence f. Then the hypersurfaces V n−1 and V n−1 are conformally equivalent. 1. In local differential geometry the rigidity theorems contain conditions under which two submanifolds of a homogeneous space can differ only by a location in the space. For hypersurfaces in a projective space, the rigidity problem was considered by G. The problem of conformal rigidity of submanifolds is also of great interest. This problem was studied by Cartan [C 17], M. do Carmo and M. Dajczer [CD 87] and R. Sacksteder [S 62] (see also the paper [Su 82] by R. Sulanke in which the author considered problems close to the rigidity problem). However, in these papers the problem of conformal rigidity was investigated in the framework of Euclidean geometry. In the current paper we present the solution of this problem in the framework of con-formal differential geometry. To this end, we introduce a conformal quadratic element and prove that if n ≥ 4 and there exists a one-to-one point correspondence of two hypersurfaces both not having umbilical points preserving this quadratic element, then the hypersurfaces are conformally equivalent. Moreover, we consider the rigidity problem not only for hyper-surfaces of a conformal space but also for hypersurfaces of a pseudoconformal space. We only assume that a hypersurface is not isotropic, i.e. its tangent subspaces are not tangent to the …