Ann Glob Anal Geom (2008) 34:287–299 DOI 10.1007/s10455-008-9112-1 ORIGINAL PAPER Contact metric manifolds with η-parallel torsion tensor Amalendu Ghosh · Ramesh Sharma · Jong Taek Cho Received: 24 July 2007 / Accepted: 18 February 2008 / Published online: 12 March 2008 © Springer Science+Business Media B.V. 2008 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 (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. Keywords η-Parallel torsion tensor · (k ,μ)-Contact manifold · Tangent sphere bundle · Ricci soliton · Sasakian manifold Mathematics Subject Classification (2000) 53C15 · 53C25 · 53C21 1 Introduction Geometries of complex and contact manifolds interact very extensively, and provide a very interesting area of research. For example, a Kaehler structure is induced on the product of a normal contact metric (Sasakian) manifold and a real line. On a general contact metric A. Ghosh (B ) Department of Mathematics, Krishnagar Government College, Krishnanagar 741101, West Bengal, India e-mail: aghosh_70@yahoo.com R. Sharma Department of Mathematics, University of New Haven, West Haven, CT 06516, USA e-mail: rsharma@newhaven.edu J. T. Cho Department of Mathematics, Chonnam National University, CNU The Institute of Basic Sciences, Gwangju 500-757, Korea e-mail: jtcho@chonnam.ac.kr 123