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Almost conformal Ricci soliton and \(\eta\)-Ricci soliton on 3-dimensional \((\epsilon, \delta)\) trans-Sasakian manifold. (English) Zbl 1399.53123

Summary: In this paper we have shown that if a 3-dimensional \((\epsilon, \delta)\) trans-Sasakian manifold admits conformal Ricci soliton \((g, V, \lambda)\) and the vector field \(V\) is point wise collinear with the unit vector field \(\xi\) then \(V\) is a constant multiple of \(\xi\). Similarly, we have proved that under the same condition an almost conformal Ricci soliton becomes conformal Ricci soliton. We have also studied \(\eta\)-Ricci soliton on \((\epsilon, \delta)\) trans-Sasakian manifold. Finally, we have shown that if a 3-dimensional \((\epsilon, \delta)\) trans-Sasakian manifold admits conformal gradient shrinking Ricci soliton then the manifold is an Einstein manifold.

MSC:

53D10 Contact manifolds (general theory)
53D15 Almost contact and almost symplectic manifolds
53A30 Conformal differential geometry (MSC2010)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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