2006/05/23 by Murata, Satoko, Umehara, Masaaki
#53A05 #53C45 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.math/0605604
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.