Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blair D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. vol. 203. Birkhäuser, Boston-Basel-Berlin (2002)
Blair D.E., Koufogiorgos T.: When is the tangent sphere bundle locally symmetric? J. Geom. 49, 55–66 (1994)
Blair D.E., Sharma R.: Generalization of Myers’ theorem on a contact manifold Illinois. J. Math. 34, 385–390 (1990)
Boyer C.P., Galicki K.: Einstein manifolds and contact geometry. Proc. Amer. Math. Soc. 129, 2419–2430 (2001)
Boyer C.P., Galicki K., Matzeu P.: On η -Einstein Sasakian geometry, preprint. arXiv:math. DG/0406627v4 (2005)
Chow, B., Knopf, D.: The Ricci flow: An introduction. Math. Surveys Monogr. 110, Amer. Math. Soc. (2004)
Gauduchon P.: La 1-forme de torsion d’une variété hermitienne compacte. Math. Ann. 267, 495–518 (1984)
Ghosh A., Koufogiorgos T., Sharma R.: Conformally flat contact metric manifolds. J. Geom 70, 66–76 (2001)
Goldberg S.I.: Integrability of almost Kaehler manifolds. Proc. Amer. Math. Soc. 21, 96–100 (1969)
Gouli-Andreou F., Tsolakidou N.: Conformally flat contact metric manifolds with Qξ = (Trl)ξ. Contributions to Algebra and Geometry 45, 103–115 (2004)
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)
Higa T.: Weyl manifolds and Einstein–Weyl manifolds. Comment. Math. Univ. St. Paul. 42, 143–160 (1993)
Matzeu P.: Some examples of Einstein–Weyl structures on almost contact manifolds. Classical Quantum Gravity 17(24), 5079–5087 (2000)
Matzeu P.: Almost contact Einstein–Weyl structures. Manuscripta Math. 108(3), 275–288 (2002)
Narita F.: Einstein–Weyl structures on almost contact metric manifolds. Tsukuba J. Math. 22, 87–98 (1998)
Perrone D.: Contact metric manifolds whose characteristic vector field is a harmonic vector field. Diff. Geom. Appl. 20, 367–378 (2004)
Pedersen H., Swann A.: Riemannian submersions, four manifolds and Einstein–Weyl geometry. Proc. London Math. Soc. 66(3), 381–399 (1993)
Sekigawa K.: On some compact Einstein almost Kaehler manifolds. J. Math. Soc. Japan 39, 677–684 (1987)
Sharma, R.: Certain results on K-contact and (κ, μ)-contact manifolds. J. Geom. (to appear)
Tanno S.: Locally symmetric K-contact Riemannian Manifolds. Proc. Japan Acad. 43, 581–583 (1967)
Tanno S.: The topology of contact Riemannian manifolds. Illinois J. Math. 12, 700–717 (1968)
Tod K.P.: Compact 3-dimensional Einstein–Weyl structures. J. London Math. Soc. 45(2), 341–351 (1992)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghosh, A. Einstein–Weyl structures on contact metric manifolds. Ann Glob Anal Geom 35, 431–441 (2009). https://doi.org/10.1007/s10455-008-9145-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-008-9145-5