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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 823809, 20 pages
Singularities of Focal Surfaces of Null Cartan Curves in Minkowski 3-Space
1School of Mathematics Science, Harbin Normal University, Harbin 150500, China
2School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
3School of Mathematics, Changchun Normal University, Changchun 130032, China
Received 28 June 2012; Accepted 28 August 2012
Academic Editor: Gabriel Turinici
Copyright © 2012 Zhigang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- J. A. Adam, “The mathematical physics of rainbows and glories,” Physics Reports, vol. 356, pp. 229–365, 2002.
- 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.
- 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.
- J. Weiss, “Bäcklund transformations, focal surfaces and the two-dimensional Toda lattice,” Physics Letters A, vol. 137, no. 7-8, pp. 365–368, 1989.
- S. Izumiya, D. Pei, and M. T. Akahashi, “Curves and surfaces in hyperbolic space,” Banach Center Publications, vol. 65, pp. 107–123, 2004.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- J. Samuel and R. Nityananda, “Transport along null curves,” Journal of Physics A, vol. 33, no. 14, pp. 2895–2905, 2000.
- G. Tian and Z. Zheng, “The second variation of a null geodesic,” Journal of Mathematical Physics, vol. 44, no. 12, pp. 5681–5687, 2003.
- 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.
- 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.
- A. Nersessian and E. Ramos, “A geometrical particle model for anyons,” Modern Physics Letters A, vol. 14, no. 29, pp. 2033–2037, 1999.
- 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.
- K. L. Duggal, “On scalar curvature in lightlike geometry,” Journal of Geometry and Physics, vol. 57, no. 2, pp. 473–481, 2007.
- W. B. Bonnor, “Null curves in a Minkowski spacetime,” Tensor (N.S.), vol. 20, pp. 229–242, 1969.
- 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.
- 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.
- J. W. Bruce and P. J. Giblin, Curves and Singularities, Cambridge University Press, Cambridge, UK, 2nd edition, 1992.
- J. Martinet, Singularities of Smooth Functions and Maps, vol. 58, Cambridge University Press, Cambridge, UK, 1982.
- G. Wassermann, “Stability of caustics,” Mathematische Annalen, vol. 216, pp. 43–50, 1975.