Bianchi surfaces whose asymptotic lines are geodesic parallels

What is it about?

Let S be a smooth surface of class C^4 in the euclidean space E^3 which is referred to its asymptotic lines. Denote the Gaussian curvature of S by K. S is said to be a Bianchi surface if K can be expressed in the asymptotic parameters (u,v) as K=-1/(U(u)+V(v))^2 where, U and V are arbitrary functions of their arguments. In this paper, it is proved that any Bianchi surface whose asymptotic lines are geodesic parallels is either a helicoid or, in particular, a surface of revolution.

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