On spacelike curves in hyperbolic space times sphere

Section: Properties Of Gauss Mapping Of Translation Surfacementioning

confidence: 99%

Caustics of translation surfaces in Euclidean 3-space

Chen¹,

Liu²,

Miao³

2017

“…Since the end of the twentieth century, semi-Euclidean geometry has been an active area of mathematical research, and it has been applied to a variety of subjects related to differential geometry and general relativity [16]. In the view of physical point, null curves and null surfaces attracted many physicists' attentions [10,5,8,17,11]. Meanwhile, the study of the differential geometry of null curves on 3-null cone has special interest in Relativity Theory.…”

Section: Preliminariesmentioning

confidence: 99%

Some new properties of null curves on 3-null cone and unit semi-Euclidean 3-spheres

Sun¹,

Pei²

2016

Self Cite

The null curves on 3-null cone have the applications in the studying of horizon types. Via the pseudo-scalar product and Frenet equations, the differential geometry of null curves on 3-null cone is obtained. In the local sense, the curvature describes the contact of submanifolds with pseudo-spheres. We introduce the geometric properties of the null curves on 3-null cone and unit semi-Euclidean 3-spheres, respectively. On the other hand, we give the existence conditions of null Bertrand curves on 3-null cone and unit semi-Euclidean 3-spheres. c ⃝2015 All rights reserved.

“…It is different from the case of Euclidean 2-sphere. We consider a null torus T 4 2 [10], and it is a section of a null cone. It is also a product space of hyperbolic plane and de Sitter sphere, that is T 4 2 = H 2 0 × S 2 1 .…”

Section: Preliminariesmentioning

confidence: 99%

Evolutes of null torus fronts

Cui¹,

Pei²,

Yu³

2015

Self Cite

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.