Skip to main content
SpringerLink
Log in

Bochner and conformal flatness of normal metric contact pairs

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

Abstract

We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner-flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kähler Vaisman manifolds.

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. Bande, G., Hadjar, A.: Contact pair structures and associated metrics. In: Differential Geometry—Proceedings of the 8th International Colloquium, pp. 266–275. World Sci. Publ., Hackensack, NJ (2009)

  2. Bande, G., Hadjar, A.: Minimality of invariant submanifolds in metric contact pair geometry. Ann. Mat. Pura Appl. doi:10.1007/s10231-014-0412-8 (to appear)

  3. Bande, G., Hadjar, A.: On the characteristic foliations of metric contact pairs. In: Harmonic Maps and Differential Geometry. Contemp. Math., vol. 542, pp. 255–259. American Mathematical Society, Providence (2011)

  4. Bande, G., Kotschick, D.: Contact pairs and locally conformally symplectic structures. In: Harmonic Maps and Differential geometry. Contemp. Math., vol. 542, pp. 85–98. American Mathematical Society, Providence (2011)

  5. Bande, G.: Formes de Contact Généralisé, Couples de Contact et Couples Contacto-symplectiques, Thèse de Doctorat. Université de Haute Alsace, Mulhouse (2000)

    Google Scholar 

  6. Bande, G., Hadjar, A.: Contact pairs. Tôhoku Math. J. 57, 247–260 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bande, G., Hadjar, A.: On normal contact pairs. Int. J. Math. 21, 737–754 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bande, G., Blair, D.E.: Symmetry in the geometry of metric contact pairs. Math. Nachr. 286, 1701–1709 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bande, G., Blair, D.E., Hadjar, A.: On the curvature of metric contact pairs. Mediterr. J. Math. 10, 989–1009 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Blair, D.E., Ludden, G.D., Yano, K.: Geometry of complex manifolds similar to the Calabi–Eckmann manifolds. J. Differ. Geom. 9, 263–274 (1974)

    MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. Blair, D.E., Martín-Molina, V.: Bochner and conformal flatness on normal complex contact metric manifolds. Ann. Glob. Anal. Geom. 39, 249–258 (2011)

    Article  MATH  Google Scholar 

  13. Bochner, S.: Curvature and Betti numbers II. Ann. Math. 50, 77–93 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kashiwada, T.: On locally conformal Kähler structures. In: New Developments in Differential Geometry (Debrecen, 1994). Math. Appl., vol. 350, pp. 225–231. Kluwer, Dordrecht (1996)

  15. Korkmaz, B.: Normality of complex contact manifolds. Rocky Mt. J. Math. 30, 1343–1380 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Tricerri, F., Vanhecke, L.: Curvature tensors on almost Hermitian manifolds. Trans. Am. Math. Soc. 267, 365–398 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  17. Vaisman, I.: Locally conformal Kähler manifolds with parallel Lee form. Rend. Mat. 12, 263–284 (1979)

    MATH  MathSciNet  Google Scholar 

  18. Yano, K.: Differential Geometry on Complex and Almost Complex Spaces. Pergamon Press, Oxford (1964)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università degli Studi di Cagliari, Via Ospedale 72, 09124, Cagliari, Italy

    Gianluca Bande

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824-1027, USA

    David E. Blair

  3. Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, 4, Rue des Frères Lumière, 68093, Mulhouse Cédex, France

    Amine Hadjar

Authors
  1. Gianluca Bande
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David E. Blair
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Amine Hadjar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gianluca Bande.

Additional information

G. Bande was supported by P.R.I.N. 2010/11, Varietà reali e complesse: geometria, topologia e analisi armonica, Italy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bande, G., Blair, D.E. & Hadjar, A. Bochner and conformal flatness of normal metric contact pairs. Ann Glob Anal Geom 48, 47–56 (2015). https://doi.org/10.1007/s10455-015-9456-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-015-9456-2

Keywords

Search

Navigation