We define framed curves (or frontals) on Euclidean 2-sphere, give a moving frame of the framed curve and define a pair of smooth functions as the geodesic curvature of a regular curve. It is quite useful for analysing curves with singular points. In general, we can not define evolutes at singular points of curves on Euclidean 2-sphere, but we can define evolutes of fronts under some conditions. Moreover, some properties of such evolutes at singular points are given.

In this paper, we introduce a one-parameter family of Legendre curves in the unit spherical bundle over the unit sphere and the curvature. We give the existence and uniqueness theorems for one-parameter families of spherical Legendre curves by using the curvatures. Then we define an envelope for the one-parameter family of Legendre curves in the unit spherical bundle. We also consider the parallel curves and evolutes of one-parameter families of Legendre curves in the unit spherical bundle and their envelopes. Moreover, we give relationships among one-parameter families of Legendre curves in the unit spherical bundle over the unit sphere and one-parameter families of Legendre curves in the unit tangent bundle over the Euclidean plane.

As a one-parameter family of singular space curves, we consider a one-parameter family of framed curves in the Euclidean space. Then we define an envelope for a one-parameter family of framed curves and investigate properties of envelopes. Especially, we concentrate on oneparameter families of framed curves in the Euclidean 3-space. As applications, we give relations among envelopes of one-parameter families of framed space curves, one-parameter families of Legendre curves and oneparameter families of spherical Legendre curves, respectively.

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.

We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves.

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes.

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