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 ΔG = f(G+C), where Δ 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 ΔG = f(G+C) for a nonzero constant vector C.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-2-4,
author = {Miekyung Choi and Dong-Soo Kim and Young Ho Kim and Dae Won Yoon},
title = {Circular cone and its Gauss map},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {203-210},
zbl = {1268.53002},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-2-4}
}
Miekyung Choi; Dong-Soo Kim; Young Ho Kim; Dae Won Yoon. Circular cone and its Gauss map. Colloquium Mathematicae, Tome 126 (2012) pp. 203-210. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-2-4/