zbMATH — the first resource for mathematics

Singularities of focal surfaces of null Cartan curves in Minkowski 3-space. (English) Zbl 1253.53005
Summary: Singularities of the focal surfaces and the binormal indicatrix associated with a null Cartan curve will be investigated in Minkowski 3-space. The relationships will be revealed between singularities of the above two subjects and differential geometric invariants of null Cartan curves; these invariants are deeply related to the order of contact of null Cartan curves with tangential planar bundle of lightcone. Finally, we give an example to illustrate our findings.

53A04 Curves in Euclidean and related spaces
Full Text: DOI
[1] J. A. Adam, “The mathematical physics of rainbows and glories,” Physics Reports, vol. 356, pp. 229-365, 2002. · Zbl 0974.78008
[2] S. Bara, Z. Jaroszewicz, A. Kolodziejczyk, and M. Sypek, “Method for scaling the output focal curves formed by computer generated zone plates,” Optics & Laser Technology, vol. 23, pp. 303-307, 1991.
[3] I. A. Popov, N. V. Sidorovsky, I. L. Veselov, and S. G. Hanson, “Statistical properties of focal plane speckle,” Optics Communications, vol. 156, pp. 16-21, 1998.
[4] J. Weiss, “Bäcklund transformations, focal surfaces and the two-dimensional Toda lattice,” Physics Letters A, vol. 137, no. 7-8, pp. 365-368, 1989.
[5] S. Izumiya, D. Pei, and M. T. Akahashi, “Curves and surfaces in hyperbolic space,” Banach Center Publications, vol. 65, pp. 107-123, 2004. · Zbl 1063.53016
[6] S. Izumiya, D. Pei, and T. Sano, “Horospherical surfaces of curves in hyperbolic space,” Publicationes Mathematicae Debrecen, vol. 64, no. 1-2, pp. 1-13, 2004. · Zbl 1067.53004
[7] S. Izumiya, D. Pei, and T. Sano, “The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space,” Glasgow Mathematical Journal, vol. 42, no. 1, pp. 75-89, 2000. · Zbl 0970.53015
[8] S. Izumiya, D. Pei, and T. Sano, “Singularities of hyperbolic Gauss maps,” Proceedings of the London Mathematical Society, vol. 86, no. 2, pp. 485-512, 2003. · Zbl 1041.58017
[9] S. Izumiya, D. H. Pei, T. Sano, and E. Torii, “Evolutes of hyperbolic plane curves,” Acta Mathematica Sinica, vol. 20, no. 3, pp. 543-550, 2004. · Zbl 1065.53022
[10] D. Pei and T. Sano, “The focal developable and the binormal indicatrix of a nonlightlike curve in Minkowski 3-space,” Tokyo Journal of Mathematics, vol. 23, no. 1, pp. 211-225, 2000. · Zbl 0978.53004
[11] K. L. Duggal, “Constant scalar curvature and warped product globally null manifolds,” Journal of Geometry and Physics, vol. 43, no. 4, pp. 327-340, 2002. · Zbl 1025.53040
[12] A. Ferrández, A. Giménez, and P. Lucas, “Geometrical particle models on 3D null curves,” Physics Letters B, vol. 543, no. 3-4, pp. 311-317, 2002. · Zbl 0997.83006
[13] A. Ferrández, A. Giménez, and P. Lucas, “Null helices in Lorentzian space forms,” International Journal of Modern Physics A, vol. 16, no. 30, pp. 4845-4863, 2001. · Zbl 1003.53052
[14] L. P. Hughston and W. T. Shaw, “Classical strings in ten dimensions,” Proceedings of the Royal Society, vol. 414, no. 1847, pp. 423-431, 1987. · Zbl 0635.53079
[15] L. P. Hughston and W. T. Shaw, “Constraint-free analysis of relativistic strings,” Classical and Quantum Gravity, vol. 5, no. 3, pp. L69-L72, 1988. · Zbl 0637.53091
[16] J. Samuel and R. Nityananda, “Transport along null curves,” Journal of Physics A, vol. 33, no. 14, pp. 2895-2905, 2000. · Zbl 1052.83502
[17] G. Tian and Z. Zheng, “The second variation of a null geodesic,” Journal of Mathematical Physics, vol. 44, no. 12, pp. 5681-5687, 2003. · Zbl 1063.53076
[18] H. Urbantke, “Local differential geometry of null curves in conformally flat space-time,” Journal of Mathematical Physics, vol. 30, no. 10, pp. 2238-2245, 1989. · Zbl 0703.53019
[19] A. Nersessian and E. Ramos, “Massive spinning particles and the geometry of null curves,” Physics Letters B, vol. 445, no. 1-2, pp. 123-128, 1998.
[20] A. Nersessian and E. Ramos, “A geometrical particle model for anyons,” Modern Physics Letters A, vol. 14, no. 29, pp. 2033-2037, 1999.
[21] K. L. Duggal, “A report on canonical null curves and screen distributions for lightlike geometry,” Acta Applicandae Mathematicae, vol. 95, no. 2, pp. 135-149, 2007. · Zbl 1117.53019
[22] K. L. Duggal, “On scalar curvature in lightlike geometry,” Journal of Geometry and Physics, vol. 57, no. 2, pp. 473-481, 2007. · Zbl 1107.53047
[23] W. B. Bonnor, “Null curves in a Minkowski spacetime,” Tensor (N.S.), vol. 20, pp. 229-242, 1969. · Zbl 0167.20001
[24] Z. Wang and D. Pei, “Singularities of ruled null surfaces of the principal normal indicatrix to a null Cartan curve in de Sitter 3-space,” Physics Letters B, vol. 689, no. 2-3, pp. 101-106, 2010.
[25] Z. Wang and D. Pei, “Null Darboux developable and pseudo-spherical Darboux image of null Cartan curve in Minkowski 3-space,” Hokkaido Mathematical Journal, vol. 40, no. 2, pp. 219-240, 2011. · Zbl 1253.53015
[26] J. W. Bruce and P. J. Giblin, Curves and Singularities, Cambridge University Press, Cambridge, UK, 2nd edition, 1992. · Zbl 0770.53002
[27] J. Martinet, Singularities of Smooth Functions and Maps, vol. 58, Cambridge University Press, Cambridge, UK, 1982. · Zbl 0522.58006
[28] G. Wassermann, “Stability of caustics,” Mathematische Annalen, vol. 216, pp. 43-50, 1975. · Zbl 0293.58008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.