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Ricci Solitons on Almost Co-Kähler Manifolds

Part of: Global differential geometry Symplectic geometry, contact geometry

Published online by Cambridge University Press:  07 December 2018

Yaning Wang
Yaning Wang*
Affiliation:
School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, P. R. China Email: wyn051@163.com
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Abstract

In this paper, we prove that if an almost co-Kähler manifold of dimension greater than three satisfying $\unicode[STIX]{x1D702}$-Einstein condition with constant coefficients is a Ricci soliton with potential vector field being of constant length, then either the manifold is Einstein or the Reeb vector field is parallel. Let $M$ be a non-co-Kähler almost co-Kähler 3-manifold such that the Reeb vector field $\unicode[STIX]{x1D709}$ is an eigenvector field of the Ricci operator. If $M$ is a Ricci soliton with transversal potential vector field, then it is locally isometric to Lie group $E(1,1)$ of rigid motions of the Minkowski 2-space.

Keywords

almost co-Kähler manifoldRicci soliton𝜂-EinsteinLie group

MSC classification

Primary: 53D15: Almost contact and almost symplectic manifolds
Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019 , pp. 912 - 922
Copyright
© Canadian Mathematical Society 2018 

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Footnotes

This work was supported by Youth Science Foundation of Henan Normal University (No. 2014QK01).

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