Einstein–Weyl structures on contact metric manifolds

Original Paper


In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W ± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying \({Q\varphi = \varphi Q}\) is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, f η) are presented.


Einstein-Weyl structure Contact metric manifold K-contact manifold Sasakian manifold 

Mathematics Subject Classification (2000)

53C15 53C25 53C21 


  1. 1.
    Blair D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. vol. 203. Birkhäuser, Boston-Basel-Berlin (2002)Google Scholar
  2. 2.
    Blair D.E., Koufogiorgos T.: When is the tangent sphere bundle locally symmetric? J. Geom. 49, 55–66 (1994)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blair D.E., Sharma R.: Generalization of Myers’ theorem on a contact manifold Illinois. J. Math. 34, 385–390 (1990)Google Scholar
  4. 4.
    Boyer C.P., Galicki K.: Einstein manifolds and contact geometry. Proc. Amer. Math. Soc. 129, 2419–2430 (2001)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Boyer C.P., Galicki K., Matzeu P.: On η -Einstein Sasakian geometry, preprint. arXiv:math. DG/0406627v4 (2005)Google Scholar
  6. 6.
    Chow, B., Knopf, D.: The Ricci flow: An introduction. Math. Surveys Monogr. 110, Amer. Math. Soc. (2004)Google Scholar
  7. 7.
    Gauduchon P.: La 1-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267, 495–518 (1984)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ghosh A., Koufogiorgos T., Sharma R.: Conformally flat contact metric manifolds. J. Geom 70, 66–76 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goldberg S.I.: Integrability of almost Kaehler manifolds. Proc. Amer. Math. Soc. 21, 96–100 (1969)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gouli-Andreou F., Tsolakidou N.: Conformally flat contact metric manifolds with Qξ  =  (Trl)ξ. Contributions to Algebra and Geometry 45, 103–115 (2004)MATHMathSciNetGoogle Scholar
  11. 11.
    Hasegawa I., Seino M.: Some remarks on Sasakian geometry-applications of Myres’ theorem and the canonical affine connection. J. Hokkaido Univ. Educ 32(section IIA), 1–7 (1981)MathSciNetGoogle Scholar
  12. 12.
    Higa T.: Weyl manifolds and Einstein–Weyl manifolds. Comment. Math. Univ. St. Paul. 42, 143–160 (1993)MATHMathSciNetGoogle Scholar
  13. 13.
    Matzeu P.: Some examples of Einstein–Weyl structures on almost contact manifolds. Classical Quantum Gravity 17(24), 5079–5087 (2000)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Matzeu P.: Almost contact Einstein–Weyl structures. Manuscripta Math. 108(3), 275–288 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Narita F.: Einstein–Weyl structures on almost contact metric manifolds. Tsukuba J. Math. 22, 87–98 (1998)MATHMathSciNetGoogle Scholar
  16. 16.
    Perrone D.: Contact metric manifolds whose characteristic vector field is a harmonic vector field. Diff. Geom. Appl. 20, 367–378 (2004)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Pedersen H., Swann A.: Riemannian submersions, four manifolds and Einstein–Weyl geometry. Proc. London Math. Soc. 66(3), 381–399 (1993)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Sekigawa K.: On some compact Einstein almost Kaehler manifolds. J. Math. Soc. Japan 39, 677–684 (1987)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Sharma, R.: Certain results on K-contact and (κ, μ)-contact manifolds. J. Geom. (to appear)Google Scholar
  20. 20.
    Tanno S.: Locally symmetric K-contact Riemannian Manifolds. Proc. Japan Acad. 43, 581–583 (1967)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Tanno S.: The topology of contact Riemannian manifolds. Illinois J. Math. 12, 700–717 (1968)MATHMathSciNetGoogle Scholar
  22. 22.
    Tod K.P.: Compact 3-dimensional Einstein–Weyl structures. J. London Math. Soc. 45(2), 341–351 (1992)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsKrishnagar Government CollegeKrishnagarIndia

Personalised recommendations