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Cotton solitons on almost coKähler 3-manifolds. (English) Zbl 1480.53098

Summary: Let \((M^3, g)\) be a three dimensional almost coKähler manifold such that the Reeb vector field \(\xi\) is an eigenvector field of the Ricci operator \(Q\), i.e. \(Q\xi =\rho\xi\), where \(\rho\) is a smooth function on \(M\). We prove that if \(g\) represents a Cotton soliton with potential vector field being collinear with \(\xi\), or a gradient Cotton soliton, then \(M\) is coKähler or locally conformally flat. Furthermore, when \(g\) represents a nontrivial Cotton soliton with potential vector field being orthogonal to \(\xi\), we prove that \(M\) is coKähler or locally isometric to one of the following Lie groups: \(E(2)\) or \(E(1,1)\) if \(\rho\) is constant along \(\xi\). Finally, for a \((\kappa,\mu, \nu)\)-almost coKähler manifold, we also consider the case when \(g\) is a nontrivial Cotton soliton with potential vector field being orthogonal to \(\xi\).

MSC:

53D15 Almost contact and almost symplectic manifolds
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53E20 Ricci flows
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