Skip to main content
SpringerLink
Log in

A classification of totally geodesic and totally umbilical Legendrian submanifolds of \((\kappa ,\mu )\)-spaces

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We present classifications of totally geodesic and totally umbilical Legendrian submanifolds of \((\kappa ,\mu )\)-spaces with Boeckx invariant \(I \le -1\). In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly construct.

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. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Mathematics 203, 2nd edn. Birkhäuser, Boston (2010)

    Book  Google Scholar 

  2. Blair, D.E., Koufogiorgos, T., Papantoniou, B.J.: Contact metric manifolds satisfying a nullity condition. Isr. J. Math. 91, 189–214 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boeckx, E.: A full classification of contact metric \((\kappa,\mu )\)-spaces. Ill. J. Math. 44, 212–219 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Cappelletti Montano, B., Di Terlizzi, L., Tripathi, M.M.: Invariant submanifolds of contact \((\kappa,\mu )\)-manifolds. Glasg. Math. J. 50, 499–507 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Interscience Tracts in Pure and Applied Mathematics 15, vol. I. Interscience, New York (1963)

    Google Scholar 

  6. Lotta, A.: Slant submanifolds in contact geometry. Bull. Math. Soc. Roum. 39, 183–198 (1996)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, c/ Tarfia s/n, 41012, Seville, Spain

    Alfonso Carriazo

  2. Departamento de Didáctica de las Matemáticas, Facultad de Ciencias de la Educación, Universidad de Sevilla, c/ Pirotecnia s/n, 41013, Seville, Spain

    Verónica Martín-Molina

  3. LAMAV, Université de Valenciennes, 59313, Valenciennes Cedex 9, France

    Luc Vrancken

  4. Departement Wiskunde, KU Leuven, Celestijnenlaan 200 B, 3001, Leuven, Belgium

    Luc Vrancken

Authors
  1. Alfonso Carriazo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Verónica Martín-Molina
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luc Vrancken
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Verónica Martín-Molina.

Additional information

The first two authors are partially supported by the MINECO-FEDER Grant MTM2014-52197-P. They are members of the IMUS (Instituto de Matemáticas de la Universidad de Sevilla), and of the PAIDI groups FQM-327 and FQM-226 (Junta de Andalucía, Spain), respectively.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Carriazo, A., Martín-Molina, V. & Vrancken, L. A classification of totally geodesic and totally umbilical Legendrian submanifolds of \((\kappa ,\mu )\)-spaces. Ann Glob Anal Geom 54, 173–185 (2018). https://doi.org/10.1007/s10455-018-9597-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-018-9597-1

Keywords

Mathematics Subject Classification

Search

Navigation