Lightcone dual surfaces and hyperbolic dual surfaces of spacelike curves in de Sitter 3-space

Volume 9, Issue 5, pp 2563--2576 http://dx.doi.org/10.22436/jnsa.009.05.54

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Authors

Haiming Liu - School of Mathematics, Mudanjiang Normal University, 157011 Mudanjiang, P. R. China. Donghe Pei - School of Mathematics and Statistics, Northeast Normal University, 130024 Changchun, P. R. China.

Abstract

In this paper, we consider the spacelike curves in de Sitter space and we investigate the singularities of lightcone dual surfaces and hyperbolic dual surfaces of these spacelike curves in the framework of the theory of Legendrian dualities between pseudo-spheres in Minkowski space. We classify the singularities of these subjects and reveal the relationships between their singularities and geometric invariants of spacelike curves under the action of the Lorentz group. As application and illustration of the main results, an example is given.

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Keywords

• De Sitter space
• Legendrian dualities
• lightcone dual surfaces.

MSC

•  32S25
•  53A35

References

• [1] J. W. Bruce, P. J. Giblin, Curves and Singularities (2nd.ed), Cambridge Univ. Press, Cambridge (1992)


• [2] T. Fusho, S. Izumiya, Lightlike surfaces of spacelike curves in de Sitter 3-space, J. Geom., 88 (2008), 19-29.


• [3] R. Hayashi, S. Izumiya, T. Sato, Duals of curves in hyperbolic space, Note Mat., 33 (2013), 97-106.


• [4] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, Mosc. Math. J., 9 (2009), 325-357.


• [5] S. Izumiya, Y. Jiang, D. Pei, Lightcone dualities for curves in the sphere, Q. J. Math., 64 (2012), 221-234.


• [6] S. Izumiya, Y. Jiang, T. Sato, Lightcone dualities for curves in the 3-sphere , J. Math. Phys., 54 (2013), 1-15.


• [7] H. L. Liu, Curves in the lightlike cone, Contrib. Algebra Geom., 45 (2004), 291-303.


• [8] H. L. Liu, Surfaces in the light cone, J. Math. Anal. Appl., 325 (2007), 1171-1181.


• [9] H. L. Liu, S. D. Jung, Hypersurfaces in lightlike cone, J. Geom. Phys., 58 (2008), 913-922.


• [10] H. L. Liu, Q. Meng, Representation formulas of curves in a 2 and 3 dimensional lightlike cone, Results Math., 59 (2011), 437-451.


• [11] B. O'Neil , Semi-Riemannian Geometry , Academic Press, London (1983)


• [12] Z. G. Wang, D. H. Pei, L. L. Kong, Gaussian surfaces and lightcone dual surfaces of null curves in a three dimensional nullcone with index 2, J. Geom. Phys., 73 (2013), 166-186.


• [13] G. Wassermann, Stability of caustics , Math. Ann., 216 (1975), 43-50.