Evolutes of null torus fronts

Abstract

The main goal of this paper is to characterize evolutes at singular points of curves in hyperbolic plane by analysing evolutes of null torus fronts. We have done some work associated with curves with singular points in Euclidean 2-sphere [H. Yu, D. Pei, X. Cui, J. Nonlinear Sci. Appl., 8 (2015), 678-686]. As a series of this work, we further discuss the relevance between singular points and geodesic vertices of curves and give different characterizations of evolutes in the three pseudo-spheres. c 2015 All rights reserved.Keywords: Evolute, null torus front, null torus framed curve, hyperbolic plane. 2010 MSC: 51B20, 53B50, 53A35. PreliminariesAs a subject closely related to nonlinear sciences, singularity theory [1,2,3,4,7] has been extensively applied in studying classifications of singularities of submanifolds in Euclidean spaces and semi-Euclidean spaces [11,12]. However, little information has been got at singular points from the view point of differential geometry. In this paper we characterize the behaviors at singular points of curves in hyperbolic plane.If a curve has singular points, we can not construct its moving frame. However, we can define a moving frame of a frontal for a framed curve in the unit tangent bundle. Along with the moving frame, we get a pair of smooth functions as the geodesic curvature of a regular curve. It is quite useful to analyse curves with singular points. Because we can get information at singular points through analysing framed curves. We have researched curves with singular points in Euclidean 2-sphere in [13]. In general, one can not define evolutes at singular points of curves on Euclidean 2-sphere, but we define evolutes of fronts under some conditions.

Evolutes of Fronts in de Sitter and Hyperbolic Spheres

“…We call the pair (m, n) a geodesic curvature of the framed curve. Also, we have ν ∧ µ = γ and γ ∧ µ = ν (for more details see [8,10]).…”

“…In this case, n(t) dose not change, but m(t) changes to −m(t). If (β, ν) is a framed immersion, we have (m(t), n(t)) = (0, 0) for each t ∈ I and call the pair (m, n) geodesic curvature of the framed curve (for more details see [8,10]).…”