Skip to main content
SpringerLink
Log in

Spaces of geodesics: products, coverings, connectedness

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We continue our study of the space of geodesics of a manifold with linear connection. We obtain sufficient conditions for a product to have a space of geodesics which is a manifold. We investigate the relationship of the space of geodesics of a covering manifold to that of the base space. We obtain sufficient conditions for a space to be geodesically connected in terms of the topology of its space of geodesics.

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. Ballmann, W., Gromov, M. and Schroeder, V.: Manifolds of Nonpositive Curvature, Birkäuser, Boston, 1985.

    Google Scholar 

  2. Beem, J. K. and Parker, P. E.: Pseudoconvexity and geodesic connectedness. Ann. Mat. Pura Appl. 155 (1989), 137–142.

    Google Scholar 

  3. Beem, J. K. and Parker, P. E.: The space of geodesic, Geom. Dedicata 38 (1991), 87–99.

    Google Scholar 

  4. Greene, R. E. and Wu, H.: Function Theory on Manifolds Which Possess a Pole, Lecture Notes in Math. 699, Springer-Verlag, New York, 1979.

    Google Scholar 

  5. Hawking, S. W. and Ellis, G. F. R.: The Large Scale Structure of Space-time. Cambridge University Press, 1973.

  6. Kobayashi, S.: Transformation Groups in Differential Geometry, Ergeb. Math. Grenz. 70, Springer-Verlag, New York, 1972.

    Google Scholar 

  7. Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Vol. I, Wiley-Interscience, New York, 1963.

    Google Scholar 

  8. Low, R. J.: The geometry of the space of null geodesics, J. Math. Phys. 30 (1989), 809–811.

    Google Scholar 

  9. Low, R. J.: Spaces of casual paths and naked singularities, Class. Quant. Gravity 7 (1990), 943–954.

    Google Scholar 

  10. Milnor, J. W. and Stasheff, J. D.: Characteristic Classes, Ann. Math. Studies 76, Princeton University Press, 1974.

  11. Palais, R. S.: A Global Formulation of the Lie Theory of Transformation Groups, Memoirs 22, Amer. Math. Soc., Providence, 1957.

    Google Scholar 

  12. Parker, P. E.: Spaces of geodesics, in L. Del Riego (ed), Aportaciones Mathemáticas, Serie: Notas de Investigación No. 8, pp. 61–73.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

    John K. Beem

  2. Department of Mathematics, Coventry University, CV1 5FB, Coventry, U.K.

    Robert J. Low

  3. Department of Mathematics, The Wichita State University, 67260-0033, Wichita, KS, USA

    Phillip E. Parker

Authors
  1. John K. Beem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Robert J. Low
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Phillip E. Parker
    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

Beem, J.K., Low, R.J. & Parker, P.E. Spaces of geodesics: products, coverings, connectedness. Geom Dedicata 59, 51–64 (1996). https://doi.org/10.1007/BF00181526

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00181526

Mathematics Subject Classification (1991)

Key words

Search

Navigation