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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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