Different solitons associated with 3-dimensional generalized Sasakian-space-forms

Abstract

In the present paper we characterize 3-dimensional generalized Sasakian-space-forms admitting some solitons such as Einstein solitons, \(\eta \)-Einstein solitons,\(\eta \)-Ricci solitons, gradient \(\eta \)-Ricci solitons and \(\eta \)-Yamabe solitons. First we show that an Einstein soliton on a 3-dimensional generalized Sasakian-space-form \(M(f_1,f_2,f_3)\) becomes a Ricci soliton and the soliton is shrinking, steady and expanding according as \((f_1 - f_3) < 0, = 0\) and \(> 0\), respectively, where \(f_1\), \(f_2\) and \(f_3\) are smooth functions. Also, we establish that if a 3-dimensional generalized Sasakian-space-form \(M(f_1,f_2,f_3)\) admits a gradient \(\eta \)-Ricci soliton with potential function f, then \(f = log(\frac{f_1-f_3}{k})^2,\) where k is a constant. Next, we prove that if a 3-dimensional generalized Sasakian-space-form \(M(f_1,f_2,f_3)\) is an \(\eta \)-Yamabe soliton, then the soliton reduces to a Yamabe soliton and the scalar curvature is constant. Finally, we construct an example which proves the existence of our results.