Abstract
We define the notions of (S 1 t × S 2 s )-nullcone Legendrian Gauss maps and S 2+ -nullcone Lagrangian Gauss maps on spacelike surfaces in anti de Sitter 4-space. We investigate the relationships between singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian singularities. By using S 2+ -nullcone Lagrangian Gauss maps, we define the notion of S 2+ -nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion of horospherical Gauss maps which have geometric properties different from those of the above Gauss maps. As a consequence, we can say that anti de Sitter space has much richer geometric properties than the other space forms such as Euclidean space, hyperbolic space, Lorentz-Minkowski space and de Sitter space.
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References
V. I. Arnold, S. M. Gusein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhäuser, Boston, 1985), Vol. 1.
T. Banchoff, T. Gaffney, and C. McCrory, Cusps of Gauss Mappings (Pitman, London, 1982), Res. Notes Math. 55.
Th. Bröcker, Differentiable Germs and Catastrophes (Cambridge Univ. Press, Cambridge, 1975), LMS Lect. Note Ser. 17.
J. W. Bruce and P. J. Giblin, Curves and Singularities, 2nd ed. (Cambridge Univ. Press, Cambridge, 1992).
R. L. Bryant, “Surfaces of Mean Curvature One in Hyperbolic Space,” in Théorie des variétés minimales et applications, Palaiseau, 1983–1984 (Soc. Math. France, Paris, 1988), Astérisque 154–155, pp. 321–347.
S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford Univ. Press, New York, 1983), Int. Ser. Monogr. Phys. 69.
C. L. Epstein, “The Hyperbolic Gauss Map and Quasiconformal Reflections,” J. Reine Angew. Math. 372, 96–135 (1986).
V. Guillemin and A. Pollack, Differential Topology (Prentice-Hall, Englewood Cliffs, NJ, 1974).
S. Izumiya, M. Kossowski, D. Pei, and M. C. Romero Fuster, “Singularities of Lightlike Hypersurfaces in Minkowski Four-Space,” Tohoku Math. J. 58, 71–88 (2006).
S. Izumiya, J. J. Nuño Ballesteros, and M. C. Romero Fuster, “Global Properties of Codimension Two Spacelike Submanifolds in Minkowski Space,” Adv. Geom., doi: 10.1515/ADVGEOM.2009.034 (2009).
S. Izumiya, D. Pei, and M. C. Romero Fuster, “The Lightcone Gauss Map of a Spacelike Surface in Minkowski 4-Space,” Asian J. Math. 8, 511–530 (2004).
S. Izumiya, D. Pei, and T. Sano, “Singularities of Hyperbolic Gauss Maps,” Proc. London Math. Soc., Ser. 3, 86, 485–512 (2003).
S. Izumiya and M. C. Romero Fuster, “The Horospherical Gauss-Bonnet Type Theorem in Hyperbolic Space,” J. Math. Soc. Japan 58, 965–984 (2006).
S. Izumiya and M. C. Romero Fuster, “The Lightlike Flat Geometry on Spacelike Submanifolds of Codimension Two in Minkowski Space,” Sel. Math. 13(1), 23–55 (2007).
M. Kossowski, “The S 2-valued Gauss Maps and Split Total Curvature of Space-like Codimension-2 Surface in Minkowski Space,” J. London Math. Soc., Ser. 2, 40, 179–192 (1989).
M. Kossowski, “The Intrinsic Conformal Structure and Gauss Map of a Light-like Hypersurface in Minkowski Space,” Trans. Am. Math. Soc. 316(1), 369–383 (1989).
J. A. Little, “On Singularities of Submanifolds of Higher Dimensional Euclidean Spaces,” Ann. Mat. Pura Appl., Ser. 4, 83, 261–335 (1969).
J. Martinet, Singularities of Smooth Functions and Maps (Cambridge Univ. Press, Cambridge, 1982), LMS Lect. Note Ser. 58.
J. N. Mather, “Stability of C ∞-Mappings. IV: Classification of Stable Germs by R-Algebras,” Publ. Math., Inst. Hautes Étud. Sci. 37, 223–248 (1969).
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W.H. Freeman and Co., San Francisco, CA, 1973).
J. A. Montaldi, “On Contact between Submanifolds,” Mich. Math. J. 33, 195–199 (1986).
J. Montaldi, “On Generic Composites of Maps,” Bull. London Math. Soc. 23, 81–85 (1991).
B. O’Neill, Semi-Riemannian Geometry (Academic, New York, 1983).
G. Wassermann, “Stability of Caustics,” Math. Ann. 216, 43–50 (1975).
H. Whitney, “On Singularities of Mappings of Euclidean Spaces. I: Mappings of the Plane into the Plane,” Ann. Math., Ser. 2, 62, 374–410 (1955).
V. M. Zakalyukin, “Lagrangian and Legendrian Singularities,” Functs. Anal. Prilozh. 10(1), 26–36 (1976) [Funct. Anal. Appl. 10, 23–31 (1976)].
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Dedicated to Vladimir Igorevich Arnold for his continuous support to the researchers of Singularity Theory
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Izumiya, S., Pei, D. & Fuster, M.d.C.R. Spacelike surfaces in anti de Sitter four-space from a contact viewpoint. Proc. Steklov Inst. Math. 267, 156–173 (2009). https://doi.org/10.1134/S0081543809040130
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DOI: https://doi.org/10.1134/S0081543809040130