Ricci Solitons in (ε,δ)-Trans-Sasakian Manifolds

Abstract

We study Ricci solitons in (ε,δ)-trans-Sasakian manifolds. It is shown that a symmetric parallel second order covariant tensor in a (ε,δ)-trans-Sasakian manifold is a constant multiple of the metric tensor. Using this it is shown that if Vg + 2S is parallel where V is a given vector field, then (g,V) is Ricci soliton. Further, by virtue of this result, Ricci solitons for 3-dimensional (ε,δ)-trans-Sasakian manifolds are obtained. Also an example of Ricci solitons in 3-dimensional (ε,δ)-trans-Sasakian manifold is provided in the region where trans-Sasakian manifold is expanding (shrinking) the Lorentzian trans-Sasakian manifold is shrinking (expanding).