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Surfaces of codimension two with zero normal torsion carrying a conjugate coordinate net in the Euclidean space. (Russian) Zbl 0671.53006
Let \(F^ n\) be a surface of class \(C^ 3\) in a Euclidean space \(E^{n+2}\), \(n\geq 2\). Suppose that \(x\in F^ n\) and \(\bar t\in T_ xF^ n\) be a tangent vector to \(F^ n\). Denote by \(E^ 3(x;\bar t,N)\subset E^{n+2}\) a 3-plane passing through the point x in the direction of the vector \(\bar t\) and the normal plane N at the point x. The plane \(E^ 3(x;\bar t,N)\) intersects \(F^ n\) along a curve \(\gamma\) which is called a normal section of \(F^ n\) at x in the direction of \(\bar t.\) The curvature \(k_ N=k_ N(x;\bar t)\) and the torsion \(\kappa_ N=\kappa_ N(x;\bar t)\) of the curve \(\gamma\) in \(E^ 3(x;\bar t,N)\) at x are called respectively the normal curvature and the normal torsion of \(F^ n\) at x in the direction \(\bar t.\)
A surface \(S^ k\subset E^{k+1}\) is said to be a generalized cyclide if its curvature lines are hyperspherical curves in \(E^{k+1}\). Suppose that \(S^{n-k}\subset E^{n-k+1}\) is another generalized cyclide. If \(F^ n=S^ k\times S^{n-k},\) we will say that \(F^ n\) is decomposed into the Riemannian product of generalized cyclides. The main result of the paper under review is the following theorem: If \(F^ n\) has a conjugate coordinate net in a neighborhood of each of its points and \(k_ N(x;\bar t)\neq 0,\quad \kappa_ N(x;\bar t)=0,\quad x\in F^ n,\quad \bar t\in T_ xF^ n,\) then \(F^ n\) belongs to a hyperplane \(E^{n+1}\subset E^{n+2}\) or it is decomposed into the Riemannian product of generalized cyclides. This result is a generalization of the result for \(F^ 2\subset E^ 4\) of the first author [Math. USSR, Sb. 35, 251-254 (1979); translation from Mat. Sb., Nov. Ser. 106(148), 589- 603 (1978; Zbl 0394.53003)].
Reviewer: V.V.Goldberg

MSC:
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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