Yamabe solitons on \(3\)-dimensional contact metric manifolds with \(Q \varphi = \varphi Q\). (English) Zbl 1422.53034
Summary: If \(M\) is a 3-dimensional contact metric manifold such that \(Q \varphi = \varphi Q\) which admits a Yamabe soliton \((g, V)\) with the flow vector field \(V\) pointwise collinear with the Reeb vector field \(\xi\), then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if \(M\) is endowed with a Yamabe soliton \((g, V)\), then either \(M\) is flat or it has constant scalar curvature and the flow vector field \(V\) is Killing. Furthermore, we show that if \(M\) is non-flat, then either \(M\) is a Sasakian manifold of constant curvature \(1\) or \(V\) is an infinitesimal automorphism of the contact metric structure on \(M\).
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 53D15 | Almost contact and almost symplectic manifolds |
| 35C08 | Soliton solutions |
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