RAJENDRA PRASAD, PANKAJ, M. M. TRIPATHI and S. S. SHUKLA
On some special type of trans-Sasakian manifolds
Received: 15 January 2008
Accepted in revised form: 27 February 2009
Mathematics Subject Classification (2000): 53B05 - 53B15
Keywords: Trans-Sasakian manifold, conharmonically flat, conformally flat, projectively flat, η-Einstein manifold, curvature tensor, Ricci-curvature, Ricci-operator.
Abstract We obtained necessary and sufficient condition for a 3-dimensional trans-Sasakian manifold of type (α,β) to be η-Einstein. In particular expressions for Ricci-tensor, curvature tensor and Ricci-operator obtained in 3-dimensional trans-Sasakian and η-Einstein trans-Sasakian manifolds. We also prove that in a three dimensional trans-Sasakian manifold of type (α,β), QΦ = ΦQ if Φ (grad α) = grad β. It is also proved that every 3-dimensional α-Sasakian, β-Kenmotsu and (α,β) trans-Sasakian manifolds, where (α,β) are constants, are always η-Einstein manifolds. Under certain conditions eigenvalues and eigenvectors are also discussed. We also obtained condition for projectively flat trans-Sasakian manifold to be Einstein.