Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

LORENTZIAN SURFACES WITH CONSTANT CURVATURES AND TRANSFORMATIONS IN THE 3-DIMENSIONAL LORENTZIAN SPACE

  • Published : 2008.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We study Lorentzian surfaces with the constant Gaussian curvatures or the constant mean curvatures in the 3-dimensional Lorentzian space and their transformations. Such surfaces are associated to the Lorentzian Grassmannian systems and some transformations on such surfaces are given by dressing actions on those systems.

Keywords

References

  1. M. Bruck, X. Du, J. Park, and C. L. Terng, The submanifold geometries associated to Grassmannian systems, Mem. Amer. Math. Soc. 155 (2002), no. 735
  2. S. S. Chern, Geometrical interpretations of the sinh-Gordon equation, Ann. Polon. Math. 39 (1980), 74-80
  3. M. Dajczer and R. Tojeiro, Commuting Codazzi tensors and the Ribaucour transformation for submanifolds, Results Math. 44 (2003), no. 3-4, 258-278 https://doi.org/10.1007/BF03322986
  4. L. P. Eisenhart, A Treatise on the Differential Geometry of curves and Surfaces, Dover, 1960
  5. J. Inoguch, Timelike Surfaces of constant mean curvature in Minkowski 3-pace, Tokyo J. Math. 21 (1998), no. 1, 141-152 https://doi.org/10.3836/tjm/1270041992
  6. J. Inoguch, Darboux transformations on timelike constant mean curvature surfaces, J. Geom. Phys. 32 (1999), no. 1, 57-78 https://doi.org/10.1016/S0393-0440(99)00011-X
  7. B. O'Neill, Semi-Riemannian Geometry, Academic Press, 1983
  8. B. Palmer, Backlund transformation for surfaces in Minkowski space, J. Math. Phys. 31 (1990), no. 12, 2872-2875 https://doi.org/10.1063/1.528939
  9. C. L. Terng, Soliton equations and differential geometry, J. Differential Geom. 45 (1997), no. 2, 407-445 https://doi.org/10.4310/jdg/1214459804
  10. C. Tian, Backlund transformations on surfaces with K = -1 in $R^{2,1}$, J. Geom. Phys. 22 (1997), no. 3, 212-218 https://doi.org/10.1016/S0393-0440(96)00036-8
  11. T. Weinstein, An Introduction to Lorentz surfaces, de Gruyter Expositions in Math., 22, Walter de Gruyter, Berlin, 1996