Pseudo-spherical normal Darboux images of curves on a timelike surface in three dimensional Lorentz–Minkowski space

Izumiya, Shyūichi; Nabarro, Ana Claudia; Sacramento, Andrea de Jesus

doi:10.1016/j.geomphys.2015.07.014

Cited by 12 publications

(14 citation statements)

References 9 publications

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“…The first task is to establish a suitable frame as the basic tool of the investigations. Extensive literature exists about the frames along curves in Eculidean space or Minkowski space 15,16,27,28 . Here, taking a different approach than done for spacelike curves lying in a nonlightlike surface, we establish a frame and call it the null Darboux frame, along the curve

γ

in the course of the study.…”

Section: Introductionmentioning

confidence: 99%

“…Geometric objects in Minkowski space, regarding singularity, have been studied extensively by, among others, the second author and by previous researchers. [14][15][16][17][18][19][20][21][22][23][24][25][26] However, to the best of the authors' knowledge, the singularities of surfaces and curves as they relate to a curve lying in a general lightlike surface have not been considered in the literature, aside from the case of lightcone. Thus, the current study hopes to serve such a need.…”

Section: Introductionmentioning

confidence: 99%

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Generalized focal surfaces of spacelike curves lying in lightlike surfaces

Liu

Wang

2020

Math Methods in App Sciences

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The main work of this paper is to investigate two kinds of generalized focal surfaces and two kinds of evolutes generated by spacelike curve γ lying in lightlike surfaces in Minkowski three‐space. Applying the method of unfolding theory in singularity theory to our study, it is shown that there exist the cuspidal edge and the swallowtail types of singularities in each of two classes of generalized focal surfaces under certain conditions; the only cusps will appear in each of evolutes. Two new geometric invariants are presented to classify the singularities of generalized focal surfaces and evolutes. Much more importantly, we reveal the correspondence among the geometric invariants, the types of singularities on generalized focal surfaces and evolutes, the singularities of two kinds of evolutes, and the contact of γ with the osculating spheres. Finally, several examples are presented to demonstrate the correctness of the theoretical results.

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γ

in the course of the study.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized focal surfaces of spacelike curves lying in lightlike surfaces

Liu

Wang

2020

Math Methods in App Sciences

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“…When we investigate a space curve lying in a surface, it is more convenient for us to use the Lorentzian Darboux frame along the curve as the basic tool than the Frenet‐Serret type frame in Lorentzian space. There are several papers about Lorentzian Darboux frame …”

Section: Introductionmentioning

confidence: 99%

“…There are several papers about Lorentzian Darboux frame. [6][7][8] It is well known that there exist three kinds of submanifolds, that is, spacelike submanifolds, timelike submanifolds, and lightlike submanifolds in Lorentz-Minkowski space. Most researchers focus on the submanifolds immersed in Lorentzian space forms with constant curvature from the viewpoint of singularity, the geometric properties of these submanifolds have been explored extensively by the other scholars and the second author.…”

Section: Introductionmentioning

confidence: 99%

Tubular surfaces of center curves on spacelike surfaces in Lorentz‐Minkowski 3‐space

Wang

Tang

2019

Math Methods in App Sciences

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In this paper, we introduce three kinds of tubular surfaces associated to original center curves γ lying in spacelike surfaces in Lorentz‐Minkowski 3‐space. It is demonstrated that these tubular surfaces can occur some singularities and the types of these singularities can be characterised by several invariants, respectively. Some interesting relations between the contacts of original curve γ with osculating model surfaces, the contacts of γ with slices, and the singularities of three kinds of surfaces are further revealed. Several examples are presented to explain the theoretical results.

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“…Os resultados desta tese foram submetidos para publicação conforme os artigos [12], [13] e [21], onde o artigo [13] já foi aceito.…”

Section: Introductionunclassified

Curvas no espaço de Minkowski

Sacramento¹

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We study in this thesis the geometry of curves in Minkowski 3-space and 4-space using singularity theory, more specifically, the contact theory. For this we study the families of height functions and of the distance square functions on the curves. The discriminant sets and bifurcation sets of these families are essential tools in our work. For curves in Minkowski 3-space, we study their focal sets and the bifurcation set of the family of the distance square functions on these curves in order to investigate what happens near the lightlike points. We also study the spherical focal sets and bifurcation sets of curves in the de Sitter space in Minkowski 3-space and 4-space. We define pseudo-spherical normal Darboux images of curves on a timelike surface in Minkowski 3-space and study the singularities and geometric properties of these normal Darboux images. Furthermore, we investigate the relation of the de Sitter (hyperbolic) normal Darboux image of a spacelike curve in S 2 1 with the lightlike surface along this spacelike curve. We define the horospherical and hyperbolic dual surfaces of spacelike curves in de Sitter space S 3 1 and study these surfaces using singularity theory technics. We give a relation between these surfaces from the view point of Legendrian dualities. Finally, we consider curves on a spacelike hypersurface in Minkowski 4-space and define the hyperbolic surface of this curve. We study the local geometry of the hyperbolic surface and of the hyperbolic curve that is defined as being the locus of singularities of the hyperbolic surface.

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Pseudo-spherical normal Darboux images of curves on a timelike surface in three dimensional Lorentz–Minkowski space

Cited by 12 publications

References 9 publications

Generalized focal surfaces of spacelike curves lying in lightlike surfaces

Generalized focal surfaces of spacelike curves lying in lightlike surfaces

Tubular surfaces of center curves on spacelike surfaces in Lorentz‐Minkowski 3‐space

Curvas no espaço de Minkowski

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