Tohoku Mathematical Journal

Singularities of tangent lightcone map of a timelike surface in Minkowski 4-space

Lingling Kong and Donghe Pei

Full-text: Open access

Abstract

In the paper, we will define tangent lightcone map, tangentlightcone curvature and tangent lightcone height function. Then we study the geometry of the timelike surfaces in Minkowski 4-space through their contact with spacelike hyperplane and give the classification of singularities of tangent lightcone map based on the Legendrian singularity theory of Arnol'd.

Article information

Source
Tohoku Math. J. (2), Volume 61, Number 4 (2009), 455-473.

Dates
First available in Project Euclid: 21 January 2010

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1264084494

Digital Object Identifier
doi:10.2748/tmj/1264084494

Mathematical Reviews number (MathSciNet)
MR2598244

Zentralblatt MATH identifier
1187.53014

Subjects
Primary: 53A35: Non-Euclidean differential geometry
Secondary: 58C27

Keywords
Timelike surface tangent lightcone map tangent lightcone height function

Citation

Kong, Lingling; Pei, Donghe. Singularities of tangent lightcone map of a timelike surface in Minkowski 4-space. Tohoku Math. J. (2) 61 (2009), no. 4, 455--473. doi:10.2748/tmj/1264084494. https://projecteuclid.org/euclid.tmj/1264084494


Export citation

References

  • V. I. Arnol'd, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps vol. I, Momogr. Math. 82, Birkhäuser Boston, Inc., Boston, MA, 1985.
  • T. Banchoff, T. Gaffney and C. McCrory, Cusps of Gauss mappings, Res. Notes in Math. 55, Pitman, London, 1982.
  • J. W. Bruce and P. J. Giblin, Generic curves and surfaces, J. London Math. Soc. 24 (1981), 555--561.
  • J. W. Bruce, On singularities, envelopes and elementary differential geometry, Math. Proc. Cambrige Philos. Soc. 89 (1981), 43--48.
  • J. W. Bruce and P. J. Giblin, Curves and singularities (second edition), Cambridge University Press, Cambridge, 1992.
  • M. Fukuda and T. Fukuda, Algebras $Q(f)$ determine the toplogical types of generic map germs, Invent. Math. 51 (1979), 231--237.
  • C. G. Gibson, K. Wirthmuller, A. A. Du Plessis and E. J. Looijenga, Topological stability of smooth mapings, Lecture Notes in Math. 552, Springer-Verlag, Berlin-New York, 1976.
  • S. Izumiya, Topological properties of Legendrian singularities, Canad. Math. Bull. 34 (1991), 357--363.
  • S. Izumiya, D. Pei and T. Sano, Singularities of hyperbolic Gauss maps, Proc. London Math. Soc. (3) 86 (2003), 485--512.
  • S. Izumiya, D. Pei and M. C. Romero-Fuster, The lightcone Gauss map of a spacelike surface in Minkowski 4-space, Asian Math. 8 (2004), 511--530.
  • L. Kong and D. Pei, Singularities of de Sitter Gauss map of timelike hypersurface in Minkowski 4-space, Sci China Ser. A 51 (2008), 241--249.
  • J. A. Little, On singularities of submanifolds of high dimensional Euclidean space, Ann. Mat. Pura Appl. (4) 83 (1969), 261--336.
  • E. J. N. Looijenga, Structural stability of smooth families of $C^\infty$-functions, Thesis, Univ. Amsterdam, 1974.
  • J. Martinet, Singularities of smooth functions and maps, London Math. Soc. Lecture Note Ser. 58, Cambridge University Press, Cambridge-New York, 1982.
  • J. N. Mather, Stability of $C^\infty $ mappings. III. Finitely determined map germs, Inst. Hautes Études Sci. Publ. Math. No. 35 (1968), 279--308.
  • J. N. Mather, Stability of $C^\infty$-mappings IV, Classification of stable germs by $\boldsybmol R$ algebras, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 223--248.
  • J. N. Mather, How to stratify mappings and jet spaces, in SingularitiÉs d'Applications Differentiables, Lecture Notes in Math. 535, Springer-Verlag, Berlin, 1976, 128--176.
  • J. A. Montaldi, On contact between submanifolds, Michigan Math. J. 33 (1986), 195--199.
  • J. A. Montaldi, On generic composites of maps, Bull. London Math. Soc. 23 (1991), 81--85.
  • B. O'Neill, Semi-Riemannian geometry, Academic Press, New York, 1983.
  • Y. H. Wan, Generic deformations of varieties, Trans. Amer. Math. Soc. 259 (1980), 107--119.
  • G. Wassermann, Stability of Caustics, Math. Ann. 216 (1975), 43--50.
  • H. Whitney, On singularities of mappings of Euclidean spaces I, Ann. of Math. 62 (1955), 374--410.
  • V. M. Zakalyukin, Lagrangian and Legendre singularities, Funct. Anal. Appl. 10 (1976), 26--36.
  • V. M. Zakalyukin, Reconstructions of fronts and caustics depending on a parameter, and versality of mappings, (Russian) Current problems in mathematics, 22, 56--93, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1983.