Circular cone and its Gauss map

Miekyung Choi¹ ; Dong-Soo Kim² ; Young Ho Kim³ ; Dae Won Yoon¹

¹ Department of Mathematics Education and RINS Gyeongsang National University Jinju 660-701, Korea
² Department of Marthematics Chonnam National University Kwangju 500-757, Korea
³ Department of Marthematics Kyungpook National University Taegu 702-701, Korea

Colloquium Mathematicum, Tome 129 (2012) no. 2, pp. 203-210

Résumé

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 $C$.

Détail

DOI : 10.4064/cm129-2-4

Keywords: family cones typical models non cylindrical ruled surfaces among circular cones unique sense their gauss map satisfies partial differential equation similar though identical characterizing so called type submanifolds specifically gauss map circular cone has vardelta where vardelta laplacian operator non zero function constant vector prove circular cones characterized being only non cylindrical ruled surfaces vardelta nonzero constant vector