Journal of the Mathematical Society of Japan

The horospherical Gauss-Bonnet type theorem in hyperbolic space

Shyuichi IZUMIYA and María del Carmen ROMERO FUSTER

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Abstract

We introduce the notion horospherical curvatures of hypersurfaces in hyperbolic space andshow that totally umbilic hypersurfaces with vanishing curvatures are only horospheres. We also show that the Gauss-Bonnet type theorem holds for the horospherical Gauss-Kronecker curvature of a closed orientable even dimensional hypersurface in hyperbolic space.

Article information

Source
J. Math. Soc. Japan, Volume 58, Number 4 (2006), 965-984.

Dates
First available in Project Euclid: 21 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1179759532

Digital Object Identifier
doi:10.2969/jmsj/1179759532

Mathematical Reviews number (MathSciNet)
MR2276176

Zentralblatt MATH identifier
1111.53042

Subjects
Primary: 53A35: Non-Euclidean differential geometry
Secondary: 53A05: Surfaces in Euclidean space 58C27

Keywords
hyperbolic space hypersurfaces hyperbolic Gauss maps horospherical geometry Gauss-Bonnet type theorem

Citation

IZUMIYA, Shyuichi; ROMERO FUSTER, María del Carmen. The horospherical Gauss-Bonnet type theorem in hyperbolic space. J. Math. Soc. Japan 58 (2006), no. 4, 965--984. doi:10.2969/jmsj/1179759532. https://projecteuclid.org/euclid.jmsj/1179759532


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References

  • V. I. Arnol'd, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, vol.,I, Birkhäuser, 1986.
  • R. L. Bryant, Surfaces of mean curvature one in hyperbolic space, In: Théorie des variétés minimales et applications (Palaiseau, 1983–1984), Astérisque, No.,154–155 (1987), 12, 321–347, 353 (1988).
  • T. E. Cecil and P. J. Ryan, Distance functions and umbilic submanifolds of hyperbolic space, Nagoya Math. J., 74 (1979), 67–75.
  • C. L. Epstein, The hyperbolic Gauss map and quasiconformal reflections, J. Reine Angew. Math., 372 (1986), 96–135.
  • C. L. Epstein, Envelopes of Horospheres and Weingarten Surfaces in Hyperbolic 3-Space, Preprint, Princeton Univ., 1984.
  • V. V. Goryunov, Projections of Generic Surfaces with Boundaries, Adv. Soviet Math., 1 (1990), 157–200.
  • M. Golubitsky and V. Guillemin, Contact equivalence for Lagrangian manifold, Adv. Math., 15 (1975), 375–387.
  • V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, 1974.
  • S. Izumiya and T. Marar, The Euler characteristic of a generic wavefront in a $3$-manifold, Proc. Amer. Math. Soc., 118 (1993), 1347–1350.
  • S. Izumiya, D. Pei, T. Sano and E. Torii, Evolutes of hyperbolic plane curves, Acta. Math. Sinica., 20 (2004), 543–550.
  • S. Izumiya, D. Pei and T. Sano, Singularities of hyperbolic Gauss maps, Proc. London Math. Soc., 86 (2003), 485–512.
  • S. Izumiya, D. Pei and M. Takahashi, Singularities of evolutes of hypersurfaces in hyperbolic space, Proc. Edinburgh Math. Soc., 47 (2004), 131–153.
  • T. Kobayashi, Null varieties for convex domains, Reports on unitary representation seminar, 6 (1986), 1–18.
  • T. Kobayashi, Asymptotic behaviour of the null variety for a convex domain in a non-positively curved space form, Journal of the Faculty of Science, University of Tokyo, 36 (1989), 389–478.
  • J. J. Nuño-Ballesteros and O. Saeki, On the number of singularities of a generic surface with boundary in a $3$-manifold, Hokkaido Math. J., 27 (1998), 517–544.
  • T. Ozawa, On the number of tritangencies of a surface in $\R^3$, In: Global Differential Geometry and Global Analysis 1984, (Ed. D. Ferus et al.), Lecture Notes in Math., 1156 Springer, 1985, 240–253.
  • S. J. Patterson and A. Perry (Appendix A by Ch. Epstein), The divisor of Selberg's zeta function for Kleinian groups, Duke Mathematical J., 106 (2001), 321–390.
  • A. Szücs, Surfaces in $\R^3$, Bull. London Math. Soc., 18 (1986), 60–66.
  • M. Umehara and K. Yamada, Surfaces of constant mean curvature $c$ in $H^3(-c^2)$ with prescribed hyperbolic Gauss map, Math. Ann., 304 (1996), 203–224.
  • I. Vaisman, A first course in differential geometry, Dekker, 1984.