Contact metric manifolds with η-parallel torsion tensor
- 231 Downloads
We show that a non-Sasakian contact metric manifold with η-parallel torsion tensor and sectional curvatures of plane sections containing the Reeb vector field different from 1 at some point, is a (k, μ)-contact manifold. In particular for the standard contact metric structure of the tangent sphere bundle the torsion tensor is η-parallel if and only if M is of constant curvature, in which case its associated pseudo-Hermitian structure is CR- integrable. Next we show that if the metric of a non-Sasakian (k, μ)-contact manifold (M, g) is a gradient Ricci soliton, then (M, g) is locally flat in dimension 3, and locally isometric to E n+1 × S n (4) in higher dimensions.
Keywordsη-Parallel torsion tensor (k, μ)-Contact manifold Tangent sphere bundle Ricci soliton Sasakian manifold
Mathematics Subject Classification (2000)53C15 53C25 53C21
Unable to display preview. Download preview PDF.
- 1.Blair, D.E. Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics 203, Birkhäuser, Boston, Basel, Berlin (2002)Google Scholar
- 9.Chern, S.S., Hamilton, R.S.: On Riemannian metrics adapted to three dimensional contact manifolds. Lecture Note in Mathematics, vol. 1111. pp. 279–308. Springer, BerlinGoogle Scholar
- 10.Chow, B., Knopf, D.: The Ricci flow: an introduction. Mathematical surveys and monographs 110. American Mathematical Society (2004)Google Scholar
- 14.Sharma, R.: Certain results on K-contact and (k, μ)-contact manifolds. J. Geom. (to appear)Google Scholar
- 17.Yano, K.: Integral formulas in Riemannian geometry. M. Dekker Inc. (1970)Google Scholar
- 18.Yano, K., Ishihara, S.: Tangent and cotangent bundles. M. Dekker Inc. (1973)Google Scholar