Journal of Mathematical Physics, Analysis, Geometry 2020, vol. 16, No 4, pp. 402-417   https://doi.org/10.15407/mag16.04.402     ( to contents , go back )

### Ricci Solitons and Certain Related Metrics on Almost Co-Kaehler Manifolds

Devaraja Mallesha Naik

Department of Mathematics, CHRIST (Deemed to be University), Bengaluru-560029, Karnataka, India
E-mail: devaraja.mallesha@christuniversity.in

V. Venkatesha

Department of Mathematics, Kuvempu University, Shankaraghatta, Karnataka 577 451, India
E-mail: vensmath@gmail.com

H. Aruna Kumara

Department of Mathematics, Kuvempu University, Shankaraghatta, Karnataka 577 451, India
E-mail: arunmathsku@gmail.com

Received November 11, 2019, revised April 1, 2020.

Abstract

In the paper, we study a Ricci soliton and a generalized $m$-quasi-Einstein metric on almost co-Kaehler manifold $M$ satisfying a nullity condition. First, we consider a non-co-Kaehler $(\kappa, \mu)$-almost co-Kaehler metric as a Ricci soliton and prove that the soliton is expanding with $\lambda=-2n\kappa$ and the soliton vector field $X$ leaves the structure tensors $\eta,\xi$ and $\varphi$ invariant. This result extends Theorem 5.1 of [32]. We construct an example to show the existence of a Ricci soliton on $M$. Finally, we prove that if $M$ is a generalized $(\kappa, \mu)$-almost co-Kaehler manifold of dimension higher than 3 such that $h\neq 0$, then the metric of $M$ can not be a generalized $m$-quasi-Einstein metric, and this recovers the recent result of Wang [37, Theorem 4.1] as a special case.

Mathematics Subject Classification 2010: 53C25, 53C15, 53D15
Key words: almost co-Kaehler manifold, Ricci soliton, generalized m- quasi-Einstein metric, (κ, μ)-nullity distribution