Global differential invariants of nondegenerate hypersurfaces

Let {gij(x)}ni,j =1 and {Lij(x)}ni,j =1 be the sets of all coefficients of the first and second fundamental forms of a hypersurface x in Rn+1. For a connected open subset U C Rn and a C???-mapping x : U + Rn+1 the hypersurface x is said to be d-nondegenerate, where d E {1, 2, ... n}, if Ldd(x) =?? 0 for all u E U. Let M(n) = {F : Rn ???+ Rn | Fx = gx + b, g E O(n), b E Rn}, where O(n) is the group of all real orthogonal n x n-matrices, and SM(n) = {F E M(n) | g E SO(n)}, where SO(n) = {g E O(n) | det(g) = 1}. In the present paper, it is proved that the set {gij(x), Ldj(x), i, j = 1,2,. .., n} is a complete system of a SM(n + 1)-invariants of a d-non-degenerate hypersurface in Rn+1. A similar result has obtained for the group M(n + 1).