Geometric invariants and focal surfaces of spacelike curves in de Sitter space from a caustic viewpoint

Haiming Liu; Jiajing Miao; Haiming Liu; Jiajing Miao

doi:10.3934/math.2021192

AIMS Mathematics

2021, Volume 6, Issue 4: 3177-3204. doi: 10.3934/math.2021192

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Research article

Geometric invariants and focal surfaces of spacelike curves in de Sitter space from a caustic viewpoint

Haiming Liu ,
Jiajing Miao ^,

School of Mathematics, Mudanjiang Normal University, Mudanjiang, 157011, China

Received: 02 October 2020 Accepted: 11 January 2021 Published: 15 January 2021
MSC : 53A25, 58K35

The focal surface of a generic space curve in Euclidean $ 3 $-space is a classical subject which is a two dimensional caustic and has Lagrangian singularities. In this paper, we define the first de Sitter focal surface and the second de Sitter focal surface of de Sitter spacelike curve and consider their singular points as an application of the theory of caustics and Legendrian dualities. The main results state that de Sitter focal surfaces can be seen as two dimensional caustics which have Lagrangian singularities. To characterize these singularities, a useful new geometric invariant $ \rho(s) $ is discovered and two dual relationships between focal surfaces and spacelike curve are given. Three examples are used to demonstrate the main results.
- invariant,
- caustic,
- focal surface,
- de Sitter space,
- singularity theory
Citation: Haiming Liu, Jiajing Miao. Geometric invariants and focal surfaces of spacelike curves in de Sitter space from a caustic viewpoint[J]. AIMS Mathematics, 2021, 6(4): 3177-3204. doi: 10.3934/math.2021192

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Abstract

The focal surface of a generic space curve in Euclidean $ 3 $-space is a classical subject which is a two dimensional caustic and has Lagrangian singularities. In this paper, we define the first de Sitter focal surface and the second de Sitter focal surface of de Sitter spacelike curve and consider their singular points as an application of the theory of caustics and Legendrian dualities. The main results state that de Sitter focal surfaces can be seen as two dimensional caustics which have Lagrangian singularities. To characterize these singularities, a useful new geometric invariant $ \rho(s) $ is discovered and two dual relationships between focal surfaces and spacelike curve are given. Three examples are used to demonstrate the main results.

References

[1]	Z. Wang, D. Pei, Singularities of ruled null surfaces of the principal normal indicatrix to a null Cartan curve in de Sitter 3-space, Phys. Lett. B, 689 (2010), 101–106.
[2]	Y. Wang, D. Pei, Singularities for normal hypersurfaces of de Sitter timelike curves in Minkowski $4$-space, Journal of Singularities, 12 (2015), 207–214.
[3]	H. Liu, G. Liu, Weingarten rotation surfaces in $3$-dimensional de Sitter space, Journal of Geometry, 79 (2004), 156–168.
[4]	H. Liu, G. Liu, Hyperbolic rotation surfaces of constant mean curvature in $3$-de Sitter space, B. Belg. Math. Soc-Sim., 7 (2000), 455–466.
[5]	H. Liu, D. Pei, Lightcone dual surfaces and hyperbolic dual surfaces of spacelike curves in de Sitter $3$-space, J. Nonlinear Sci. Appl., 9 (2016), 2563–2576.
[6]	J. Miao, H. Liu, Caustics of de Sitter spacelike curves in Minkowski 3-space, J. Nonlinear Sci. Appl., 9 (2016), 4286–4294.
[7]	S. Izumiya, A. Nabarro, A. D. J. Sacramento, Horospherical and hyperbolic dual surfaces of spacelike curves in de Sitter space, Journal of Singularities, 16 (2017), 180–193.
[8]	Z. Wang, M. He, Singularities of dual hypersurfaces and hyperbolic focal surfaces along spacelike curves in light cone in Minkowski $5$-space, Mediterr. J. Math., 16 (2019), 96.
[9]	D. Pei, Singularities of horospherical hypersurfaces of curves in hyperbolic 4-space, J. Aust. Math. Soc., 91 (2011), 89–101.
[10]	R. Hayashi, S. Izumiya, T. Sato, Focal surfaces and evolutes of curves in hyperbolic space, Commun. Korean Math. Soc., 32 (2017), 147–163.
[11]	L. Chen, S. Izumiya, A mandala of Legendrian dualities for pseudo-spheres in semi-Euclidean space, P. Jpn. Acad. A-Math., 85 (2009), 49–54.
[12]	S. Izumiya, M. D. C. R. Fuster, M. A. S. Ruas, F. Tari, Differential geometry from a singularity theory viewpoint, 1 Eds., Hackensack: World Scientific Publishing Co. Pte. Ltd., 2016.
[13]	J. W. Bruce, P. J. Giblin, Curves and Singularities, 2 Eds., Cambridge: Cambridge Press, 1992.

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