A uniqueness theorem for convex surfaces with no umbilical points and interrelated principal curvatures
type: Item
instance of: scholarly article
title: A uniqueness theorem for convex surfaces with no umbilical points and interrelated principal curvatures
zbMATH Open document ID: 0874.53004
DOI: 10.1007/BF02110734
published in: Siberian Mathematical Journal
publication date: 1997-11-11T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53A05
zbMATH DE Number: 1023917
zbMATH Keywords: convex surfaces
zbMATH Keywords: right circular cylinder
zbMATH Keywords: principal curvatures
reviewed by: T. Koufogiorgos
MaRDI profile type: Publication
cites work: Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Meinem Lehrer Erhard Schmidt in Verehrung und Freundschaft zum 75. Geburtstag gewidmet.
cites work: A uniqueness theorem for a surface with principal curvatures connected by the relation \((1-k_ 1d)(1-k_ 2d)=-1\)
cites work: On conditions for existence of umbilical points on a convex surface
author: V. A. Toponogov
description: scientific article; zbMATH DE number 1023917