Circular cone and its Gauss map
Volume 129 / 2012
Colloquium Mathematicum 129 (2012), 203-210
MSC: Primary 53B25; Secondary 53C40.
DOI: 10.4064/cm129-2-4
Abstract
The family of cones is one of typical models of non-cylindrical ruled surfaces. Among them, the circular cones are unique in the sense that their Gauss map satisfies a partial differential equation similar, though not identical, to one characterizing the so-called 1-type submanifolds. Specifically, for the Gauss map $G$ of a circular cone, one has $\varDelta G = f(G+C)$, where $\varDelta $ is the Laplacian operator, $f$ is a non-zero function and $C$ is a constant vector. We prove that circular cones are characterized by being the only non-cylindrical ruled surfaces with $\varDelta G = f(G+C)$ for a nonzero constant vector $C$.
