A uniqueness theorem for a surface with principal curvatures connected by the relation \((1-k_ 1d)(1-k_ 2d)=-1\)
Let \(F\) be an oriented complete regular surface in three-dimensional Euclidean space \(E^ 3\). The main result of this paper is as follows: If the surface \(F\) is analytic and there exists a continuous field \(n(p)\) of normals to \(F\) such that the principal curvatures \(k_ 1 (p)\) and \(k_ 2 (p)\) with signs determined by the field \(n(p)\) obey the condition \((1- k_ 1 d) (1- k_ 2 d)=-1\), then \(F\) is a circular cylinder of radius \(d/2\).
scientific article; zbMATH DE number 652188