Horospherical and hyperbolic dual surfaces of spacelike curves in de Sitter space

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

doi:10.5427/jsing.2017.16h

Cited by 5 publications

(14 citation statements)

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“…In this section, we give the basic notions and familiar results in Lorentzian geometry which we need in this paper (for more details, see [7][8][9][10][11][12]).…”

Section: Preliminariesmentioning

confidence: 99%

On involute-evolute curve couple in the hyperbolic and de Sitter spaces

2019

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“…In this section, we give the basic notions and familiar results in Lorentzian geometry which we need in this paper (for more details, see [7][8][9][10][11][12]).…”

Section: Preliminariesmentioning

confidence: 99%

On involute-evolute curve couple in the hyperbolic and de Sitter spaces

2019

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“…The Minkowski space R 4 1 is the vector space R 4 endowsed with the pseudo-scalar product x, y = −x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3 , for any x = (x 0 , x 1 , x 2 , x 3 ) and y = (y 0 , y 1 , y 2 , y 3 ) in R 4 1 (see, e.g., [9]). We say that a non-zero vector x ∈ R 4 1 is spacelike if x, x > 0, lightlike if x, x = 0 and timelike if x, x < 0, respectively.…”

Section: Preliminariesmentioning

confidence: 99%

“…We consider a spacelike embedding X : U → R 4 1 from an open subset U ⊂ R 3 and identify M and U through the embedding X, where R 4 1 is the Minkowski 4-space. For a curve γ : I → M with nowhere vanishing curvature, we define a hyperbolic surface in hyperbolic space H 3 (−1) and a de Sitter surface in the de Sitter space S 3 1 associated to the curve γ.…”

Section: Introductionmentioning

confidence: 99%

“…When γ is not part of a slice surface, we characterize the contact of γ with a slice surface using the singularity types of its hyperbolic surface (Proposition 3.7 ) and the singularity types of its de Sitter surface (Proposition 5.7). In Sections 4 and 6, we consider examples of curves on spacelike hypersurface in R 4 1 and we obtain the surfaces studied in [3].…”

Section: Introductionmentioning

confidence: 99%

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Curves in a spacelike hypersurface in the Minkowski space-time

Izumiya,

Nabarro,

Sacramento

2019

Preprint

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We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface M in the Minkowski 4-space. These surfaces are respectively located in the hyperbolic 3-space and in the de Sitter 3-space. We use techniques of the theory of singularities in order to describe the generic shape of these surfaces and of their singular value sets. We also investigate geometric meanings of those singularities.

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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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Horospherical and hyperbolic dual surfaces of spacelike curves in de Sitter space

Cited by 5 publications

References 11 publications

On involute-evolute curve couple in the hyperbolic and de Sitter spaces

On involute-evolute curve couple in the hyperbolic and de Sitter spaces

Curves in a spacelike hypersurface in the Minkowski space-time

Curvas no espaço de Minkowski

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