A uniqueness theorem for convex surfaces with no umbilical points and interrelated principal curvatures
The author obtains the following characterization of the right circular cylinder. Let \(\Phi\) be a complete, convex surface of class \(C^3\) in three-dimensional Euclidean space and \(f(x,y)\) a differentiable function defined in the domain \([0,c)\times [a,b)\), such that (1) at the points of the set \(f(x,y)=0\), either \(\frac{\partial f}{\partial x}\neq 0\) or \(\frac{\partial f}{\partial y}\neq 0\), (2) \(f(x,x)\neq 0\) for all \(x\in[0,c)\). If the principal curvatures \(k_1\) and \(k_2\) of \(\Phi\) are connected by the equation \(f(k_1,k_2)=0\), then the surface \(\Phi\) is a right circular cylinder.
scientific article; zbMATH DE number 1023917