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November 2021 Three dimensional contact metric manifolds with Cotton solitons
Xiaomin Chen
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Hiroshima Math. J. 51(3): 275-299 (November 2021). DOI: 10.32917/h2020064

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

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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

Information

Received: 11 July 2020; Revised: 7 April 2021; Published: November 2021
First available in Project Euclid: 1 December 2021

Digital Object Identifier: 10.32917/h2020064

Subjects:
Primary: 53C21
Secondary: 53C15

Keywords: contact metric $(\kappa,\mu,\nu)$-manifold , Contact metric manifold , Cotton soliton , Sasakian manifold

Rights: Copyright © 2021 Hiroshima University, Mathematics Program

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Vol.51 • No. 3 • November 2021
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