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Annals of Global Analysis and Geometry

, Volume 34, Issue 3, pp 287–299 | Cite as

Contact metric manifolds with η-parallel torsion tensor

  • Amalendu GhoshEmail author
  • Ramesh Sharma
  • Jong Taek Cho
Original Paper

Abstract

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 (kμ)-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.

Keywords

η-Parallel torsion tensor (kμ)-Contact manifold Tangent sphere bundle Ricci soliton Sasakian manifold 

Mathematics Subject Classification (2000)

53C15 53C25 53C21 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsKrishnagar Government CollegeKrishnanagarIndia
  2. 2.Department of MathematicsUniversity of New HavenWest HavenUSA
  3. 3.Department of MathematicsChonnam National University, CNU The Institute of Basic SciencesGwangjuKorea

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