Flat surfaces with singularities in euclidean 3-space

Satoko Murata Kyoto Municipal Saikyo Senior High School Masaaki Umehara Osaka University

Differential Geometry mathscidoc:1609.10118

Journal of Differential Geometry, 82, (2), 279-316, 2009
It is classically known that complete flat (that is, zero Gaussian curvature) surfaces in Euclidean 3-space R3 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  is globally defined on the surface and can be smoothly extended across the singular set, f is called a frontal. In addition, if the pair (f, ) defines an immersion into R3×S2, f is called a front. A front f is called flat if the Gauss map degenerates everywhere. The parallel surfaces and the caustic (i.e. focal surface) of a flat front f are also flat fronts. In this paper, we generalize the classical notion of completeness to flat fronts, and give a representation formula for a flat front which has a non-empty compact singular set and whose ends are all immersed and complete. As an application, we show that such a flat front has properly embedded ends if and only if its Gauss map 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.
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  author={Satoko Murata, and Masaaki Umehara},
  booktitle={Journal of Differential Geometry},
Satoko Murata, and Masaaki Umehara. FLAT SURFACES WITH SINGULARITIES IN EUCLIDEAN 3-SPACE. 2009. Vol. 82. In Journal of Differential Geometry. pp.279-316. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911183528926695775.
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