Global differential invariants of nondegenerate hypersurfaces (original) (raw)

Turkish Journal of Mathematics

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 R n+1. For a connected open subset U ⊂ R n and a C ∞-mapping x : U → R n+1 the hypersurface x is said to be d-nondegenerate, where d ∈ {1, 2,. .. n} , if L dd (x) ̸ = 0 for all u ∈ U. Let M (n) = {F : R n −→ R n | F x = gx + b, g ∈ O(n), b ∈ R n } , 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), L dj (x), i, j = 1, 2,. .. , n} is a complete system of a SM (n + 1)-invariants of a d-non-degenerate hypersurface in R n+1. A similar result has obtained for the group M (n + 1) .