Complete systems of invariants of a hyper-surface in a Euclidean space

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Let M(n+1) be the group of all isometric transformations of an n+1-dimensional Euclidean space Rn+1, SM(n + 1) is the group of all Euclidean motions of Rn+1, U is the domain f(u1)2 + · · · + (un)2 < 1g in Rn and x(u) is an U-hyper-surface (that is a C1-mapping x : U ! Rn+1) in Rn+1. Let x(u) be a regular hyper-surface, I(x) = Pn i;j=1 gij(x)duiduj and II(x) = Pn i;j=1 Lij(x)duiduj are the first and second fundamental forms of x. Put K = f(i; j) : 1 ≤ i ≤ j ≤ ng. We consider the following problem: find a proper (minimal, it is desirable) subset T of K such that equalities gij(x)(u) = gij(y)(u); Lij(x)(u) = Lij(y)(u) for all (i; j) 2 T and u 2 U implies an existence of F 2 SM(n + 1) such that y(u) = Fx(u) for all u 2 U. For a solution of this problem, we have obtained a generating system of the differential field of all G-invariant differential rational functions of a hyper-surface in Rn+1 for groups G = M(n + 1); SM(n + 1). In our result, jTj = n(n+1) 2 + n, where jTj is the number of elements of T.