Flat Surfaces with singularities in Euclidean 3-space
| dc.creator | Murata, Satoko | |
| dc.creator | Umehara, Masaaki | |
| dc.date | 2006-05-23 | |
| dc.date | 2008-12-25 | |
| dc.date.accessioned | 2026-07-07T12:22:02Z | |
| dc.date.available | 2026-07-07T12:22:02Z | |
| dc.description | It is classically known that complete flat surfaces in Euclidean 3-space are cylinders over space curves. This implies that the study of global behaviour of flat surfaces requires the study of singular points as well. If a flat surface $f$ admits singularities but its Gauss map $ν$ can be smoothly extended across the singular set, $f$ is called a frontal. In addition, if the pair $(f,ν)$ gives an immersion, $f$ is called a front. A front $f$ is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the focal surface of a flat front $f$ are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fonts, and give a representation formula for complete flat fronts. As an application, we show that a complete flat front has properly embedded ends if and only if its Gauss image is a convex curve. Moreover, we show the existence of at least four singular points other than cuspidal edges on such a flat front with embedded ends, which is a variant of the classical four vertex theorem for convex plane curves. | |
| dc.description | 31-pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/math/0605604 | |
| dc.identifier | http://arxiv.org/abs/math/0605604 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/213518 | |
| dc.subject | Differential Geometry | |
| dc.subject | 53A05; 53C45 | |
| dc.title | Flat Surfaces with singularities in Euclidean 3-space | |
| dc.type | text |