Skip to main content
SpringerLink
Log in

New Properties on Normalized Null Hypersurfaces

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

A Correction to this article was published on 01 November 2018

This article has been updated

Abstract

Rigging technique introduced in Gutiérrez and Olea (Math Nachr 289:1219–1236, 2016) is a convenient way to address the study of null hypersurfaces. It offers, in addition, the extra benefit of inducing a Riemannian structure on the null hypersurface which is used to study geometric and topological properties on it. In this paper, we develop this technique showing new properties and applications. We first discuss the very existence of the rigging fields under prescribed geometric and topological constraints. We consider the completeness of the induced rigged Riemannian structure. This is potentially important, because it allows to use most of the usual Riemannian techniques.

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

Change history

  • 01 November 2018

    There is a mistake in the last subsection (5.3 Totally geodesic null hypersurfaces in Robertson–Walker spaces) of the paper [1]

References

  1. Atindogbe, C., Berard-Bergery, L.: Distinguished normalization on non-minimal null hypersurfaces. Math. Sci. Appl. E Notes 1, 18–35 (2013)

    MATH  Google Scholar 

  2. Atindogbe, C.: Blaschke type normalization on lightLike Hypersurfaces. J. Math. Phys. Anal. Geom. 6, 362–382 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Atindogbe, C., Ezin, J.-P., Tossa, J.: Pseudo-inversion of degenerate metrics. Int. J. Math. Math. Sci. 55, 3479–3501 (2003)

    Article  Google Scholar 

  4. Atindogbe, C., Duggal, K.L.: Conformal screen on lightlike hypersurfaces. Int. J. Pure Appl. Math. 11, 421–441 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Atindogbe, C., Fosting, H.T.: Newton transformations on null hypersurfaces. Commun. Math. 23, 57–83 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Bishop, R.L., O’neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    Article  MathSciNet  Google Scholar 

  7. Johnson, D.L., Whitt, L.B.: Totally geodesic foliations. J. Differ. Geom. 15, 225–235 (1980)

    Article  MathSciNet  Google Scholar 

  8. Galloway, G.J.: Maximum principles for null hypersurfaces and null splitting theorems. Ann. Henri Poincaré 1, 543–567 (2000)

    Article  MathSciNet  Google Scholar 

  9. Gálvez, José A.: Surfaces of constant curvature in 3-dimensional space forms. Mat. Contemp. 37, 1–42 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Hilbert, D.: Über Flachen von constanter Gausscher Krummung, (German) (On surfaces of constant Gaussian curvature). Trans. Am. Math. Soc. 2, 87–99 (1901)

    Google Scholar 

  11. Duggal, K.L., Bejancu, A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Kluwer Academic, Dordrecht (1996)

    Book  Google Scholar 

  12. Fernández-López, M., Garcia-Rio, E., Kupeli, D.N., Ünal, B.: A curvature condition for a twisted product to be a warped product. Manuscr. Math. 106, 213–217 (2001)

    Article  MathSciNet  Google Scholar 

  13. Fisher, A.E.: Riemannian submersions and the regular interval theorem of Morse theory. Ann. Global Anal. Geom. 14, 263–300 (1996)

    Article  MathSciNet  Google Scholar 

  14. Gordon, W.B.: An analytical criterion for the completness of Riemannian manifolds. Proc. Am. Math. Soc. 37, 221–225 (1973)

    Article  Google Scholar 

  15. Gordon, W.B.: Corrections to “An Analytical Criterion for the Completeness of Riemannian Manifolds”. Proc. Am. Math. Soc. 45, 130–131 (1974)

    MATH  Google Scholar 

  16. Dajczer, M.: Submanifolds and Isometric Immersions, Math. Lectures Ser. vol. 13. Publish or Perish Inc. (1990)

  17. Hadamard, J.: Les surfaces à courbures opposées et leurs lignes géodésiques. J. Math. Pures Appl. 4, 27–73 (1896)

    MATH  Google Scholar 

  18. Gutiérrez, M., Olea, B.: Global decomposition of a manifold as a generalized Robertson–Walker space. Differ. Geom. Appl. 27, 146–156 (2009)

    Article  MathSciNet  Google Scholar 

  19. Gutiérrez, M., Olea, B.: Splitting theorems in presence of an irrotational vector field (2003). arXiv:math/0306343

  20. Gutiérrez, M., Olea, B.: Induced Riemannian structures on null hypersurfaces. Math. Nachr. 289, 1219–1236 (2016)

    Article  MathSciNet  Google Scholar 

  21. Thorbegsson, G.: Closed geodesic on non compact Riemannian manifolds. Math. Z. 159, 249–258 (1978)

    Article  MathSciNet  Google Scholar 

  22. Kupeli, D.N.: Degenerate submanifolds in semi-Riemannian geometry. Geom. Dedicata 24, 337–361 (1987)

    Article  MathSciNet  Google Scholar 

  23. Sakai, T.: Warped products and Riemannian manifolds admitting a function whose gradient is of constant norm. Math. J. Okayama Univ. 39, 165–185 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Sakai, T.: On Riemannian manifolds admitting a function whose gradient is of constant norm. Kodai Math. J. 19, 39–51 (1996)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The second and third authors have been partially supported by an FEDER-MTM2016-78647-P grant. The third author has been partially supported by a CEA-SMA grant.

Author information

Authors and Affiliations

  1. Institut de Mathématiques et de Sciences Physiques (IMSP), Université d’Abomey-Calavi, Abomey-Calavi, Bénin

    Cyriaque Atindogbe & Raymond Hounnonkpe

  2. Universidad de Málaga, Málaga, Spain

    Manuel Gutiérrez

Authors
  1. Cyriaque Atindogbe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manuel Gutiérrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Raymond Hounnonkpe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel Gutiérrez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Atindogbe, C., Gutiérrez, M. & Hounnonkpe, R. New Properties on Normalized Null Hypersurfaces. Mediterr. J. Math. 15, 166 (2018). https://doi.org/10.1007/s00009-018-1210-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-018-1210-0

Mathematics Subject Classification

Keywords

Search

Navigation