A note on quasi-Yamabe solitons on contact metric manifolds. (English) Zbl 1435.53035
Summary: We prove that a contact metric manifold does not admit a proper quasi-Yamabe soliton \((M, g,\xi ,\lambda ,\mu )\). Next we prove that if a contact metric manifold admits a quasi-Yamabe soliton \((M, g, V, \lambda , \mu )\) whose soliton field is pointwise collinear with the Reeb vector field, then the scalar curvature is constant, and the quasi-Yamabe soliton reduces to Yamabe soliton. Finally, it is shown that if a contact metric manifold admits a quasi-Yamabe soliton whose soliton field is the V-Ric vector field, then the Ricci operator \(Q\) commutes with the \((1, 1)\) tensor \(\phi \). As a consequence of the main result we obtain several corollaries.
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53D10 | Contact manifolds (general theory) |
| 53D15 | Almost contact and almost symplectic manifolds |
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