Abstract
In this article we study a three dimensional contact metric manifold $M^3$ with Cotton solitons. We mainly consider two classes of contact metric manifolds admitting Cotton solitons. Firstly, we study a contact metric manifold with $Q\xi = \rho\xi$, where $\rho$ is a smooth function on $M$ constant along Reeb vector field $\xi$ and prove that it is Sasakian or has constant sectional curvature 0 or 1 if the potential vector field of Cotton soliton is collinear with $\xi$ or is a gradient vector field. Moreover, if $\rho$ is constant we prove that such a contact metric manifold is Sasakian, flat or locally isometric to one of the following Lie groups: $SU(2)$ or $SO(3)$ if it admits a Cotton soliton with the potential vector field being orthogonal to Reeb vector field $\xi$. Secondly, it is proved that a $(\kappa,\mu,\nu)$-contact metric manifold admitting a Cotton soliton with the potential vector field being Reeb vector field is Sasakian. Furthermore, if the potential vector field is a gradient vector field, we prove that $M$ is Sasakian, flat, a contact metric $(0,−4)$-space or a contact metric $(\kappa,0)$-space with $\kappa \lt 1$ and $\kappa\neq0$. For the potential vector field being orthogonal to $\xi$, if $\nu$ is constant we prove that $M$ is either Sasakian, or a $(\kappa,\mu)$-contact metric space.
Funding Statement
The author is supported by Beijing Natural Science Foundation
(Grant No. 1194025) and supported partially by Science Foundation of China
University of Petroleum-Beijing (No. 2462020XKJS02, No.
2462020YXZZ004).
Acknowledgement
The author would like to thank the referee for the comments.
Citation
Xiaomin Chen. "Three dimensional contact metric manifolds with Cotton solitons." Hiroshima Math. J. 51 (3) 275 - 299, November 2021. https://doi.org/10.32917/h2020064
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