Global differential invariants of nondegenerate hypersurfaces.

Let {gij(x)}n i,j=1 and {Lij(x)}n i,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 Rn and a C 8-mapping x : U Rn+1 the hypersurface x is said to be d-nondegenerate, where d {1, 2, . . . n}, if Ldd(x) 1= 0 for all u U . Let M(n) = {F : Rn -1 Rn | Fx = gx + b, g O(n), b Rn}, where O(n) is the group of all real orthogonal n × n-matrices, and SM(n) = {F M(n) | g SO(n)}, where SO(n) = {g 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). [ABSTRACT FROM AUTHOR]