Skip to main content
SpringerLink
Log in

A Parametrization of Isometric Immersions between Hyperbolic Spaces

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We parametrize the space of isometric immersions of the hyperbolic n -space into the hyperbolic ( n +1)-space by a family of properly chosen (at most) countable n -tuples of real-valued functions. This is an answer to the problem posed by Nomizu in 1973. We also construct a one-parameter family of deformable isometric immersions from the quotient spaces of the hyperbolic 2-plane modulo and some totally discontinuous subgroups of isometries.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abe, K. and Haas, A.: Isometric immersions of H n into H n+1, in: Differential Geometry: Riemannian Geometry (Los Angeles 1990), Proc. Symp. Pure Math. 23(3), Amer. Math. Soc., Providence, RI, 1993, pp. 23–30.

    Google Scholar 

  2. Alexander, S. and Portnoy, E.: Cylindricity of isometric immersions between hyperbolic spaces, Trans. Amer. Math. Soc. 237 (1978), 311–329.

    Google Scholar 

  3. Cartan, É.: Sur les variétés de courbure constante d'un espace euclidien ou non euclidien, Bull. Soc. Math. France 47 (1919), 125–160; 48 (1920), 132–208.

    Google Scholar 

  4. do Carmo, M.: Riemannian Geometry, Birkhäuser, Boston, Basel, 1992.

    Google Scholar 

  5. do Carmo, M. and Dajczer, M.: Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277 (1983), 685–709.

    Google Scholar 

  6. Dajczer et al., M.:Submanifolds and Isometric Immersions, Publish or Perish, Houston, Texas, 1990.

    Google Scholar 

  7. Ferus, D.: On isometric immersions between hyperbolic spaces, Math. Ann. 205 (1973), 193–200.

    Google Scholar 

  8. Helgason, S.: Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962.

    Google Scholar 

  9. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. 2, Wiley-Interscience, New York, 1969.

    Google Scholar 

  10. Mori, H.: Stable complete constant mean curvature surfaces in R 3 and H 3, Trans. Amer. Math. Soc. 278 (1983), 671–687.

    Google Scholar 

  11. Nomizu, K.: Isometric immersions of the hyperbolic plane into the hyperbolic space, Math. Ann. 205 (1973), 181–192.

    Google Scholar 

  12. O'Neill, B. and Stiel, E.: Isometric immersions of constant curvature manifolds, Michigan Math. J. 10 (1963), 335–339.

    Google Scholar 

  13. Sneddon, I. N.: Elements of Partial Differential Equations, McGraw-Hill, New York, 1957.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, U-9, MSB 111, 196, Auditorium Road, Storrs, Connecticut, 06269, U.S.A.

    K. Abe

  2. Department of Mathematics, Joetsu University of Education, 943, Joetsu, Niigata Pref., Japan e-mail

    H. Mori

  3. Department of Mathematics, Ichinoseki National College of Technology, 021, Ichinoseki, Iwate Pref., Japan

    H. Takahashi

Authors
  1. K. Abe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Mori
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. H. Takahashi
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abe, K., Mori, H. & Takahashi, H. A Parametrization of Isometric Immersions between Hyperbolic Spaces. Geometriae Dedicata 65, 31–46 (1997). https://doi.org/10.1023/A:1004960317128

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004960317128

Search

Navigation