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Gradient $ρ$ $ρ$ -Einstein solitons on almost Co-Kähler manifolds

Department of Mathematics, University of Kalyani, Kalyani 741235, West Bengal, India

Corresponding author.

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Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India

E-mail Address: uc_de@yahoo.com

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https://doi.org/10.1142/S0219887823300027Cited by:1 (Source: Crossref)

Abstract

The aim of this paper is to characterize almost co-Kähler manifolds and co-Kähler three-manifolds whose metrices are the gradient $ρ$ $ρ$ -Einstein solitons. At first we prove that a proper $(˜ κ, ˜ μ)$ $(\tilde{κ}, \tilde{μ})$ -almost co-Kähler manifold with $˜ κ < 0$ $\tilde{κ} < 0$ does not admit gradient $ρ$ $ρ$ -Einstein soliton. It is also shown that if a proper $θ$ $𝜃$ -Einstein almost co-Kähler manifold with constant coefficients admits a gradient $ρ$ $ρ$ -Einstein soliton, then either the manifold is a $K$ $K$ -almost co-Kähler manifold or the soliton is trivial. Next, we prove that in case of co-Kähler three-manifold the manifold is of constant scalar curvature. Moreover, either the manifold is flat or the gradient of the potential function is collinear with the Reeb vector field $ξ$ $ξ$ . Finally, we construct two examples to illustrate our results.

Keywords:

AMSC: 53C25, 53C15

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