On analog of Darboux surfaces in many-dimensional Euclidean spaces

The Darboux tensor, symmetric covariant three-valent tensor Θ, was determined on two-dimensional surfaces with nonzero Gaussian curvature К ≠ 0 in Euclidean space E 3. The term Θ ≡ 0 is the characteristic condition of Daroux surfaces in E 3. The symmetric covariant three-valent tensor Θ( n) is determined on hypersurfaces F n (n ≥ 2) with nonzero Gaussian curvature К ≠ 0 in Euclidean space E n+1. If n = 2 then tensor Θ( n) is coincided with the Darboux tensor Θ: Θ( 2) ≡ Θ. Let V( n) be a set of hypersurfaces F n (n ≥ 2) with nonzero Gaussian curvature К = 0 in Euclidean spaces E n+1, on which the following condition holds Θ( n) ≡ 0. The set D( 2) becomes exhausted by Daroux surfaces in E 3. The properties of hypersurfaces F n ⊂ E n+1 from the set D( n) for n ≥ 2 are studied in this article. The necessary and sufficient conditions, for which hypersurface F n with nonzero Gaussian curvature К ≠ 0 in Euclidean space E n+1 belongs to the set D( n) (n ≥ 2), are derived. It was proved that hypersurface F n ⊂ E n+1 with nonzero Gaussian curvature К ≠ 0 belongs to the set D( n) (n ≥ 2) if and only if there exist coordinates of curvature (u 1,...,u n), in neighborhood O(x) ⊂ F n of every point x ∈F n, such that the following conditions hold: where k 1,..., k n are the principal curvatures F n, К = k 1 k 2...k n is Gaussian curvature of F n, ψ(і)(u i) ≠ 0, i = 1¯,n¯, are certain functions. It was proved that every cyclic recurrent hypersurface F n ⊂ E n+1 with nonzero Gaussian curvature К ≠ 0 belongs to the set D( n) (n ≥ 2). The characteristic property of hypersphere S n ⊂ E n+1 was derived. It was proved that connected complete hypersurface F n of constant positive Gaussian curvature К = const > 0 in Euclidean space E n+1, belonging to the set D( n) (n ≥ 2), is sphere S n ⊂ E n+1.