Almost \(\ast\)-\(\eta\)-Ricci solitons on Kenmotsu pseudo-Riemannian manifolds
In almost contact pseudo-Riemannian geometry, the \(\ast\)-Ricci tensor is the analogue to the usual Ricci tensor in pseudo-Riemannian geometry. The notion of the \(\ast\)-Ricci soliton is obtained by replacing of the Ricci tensor with the \(\ast\)-Ricci tensor in the classical definition of Ricci soliton: \[ L_Vg+2\mathrm{Ric}=2\lambda g. \] Some important results related to \(\ast\)-Ricci solitons are presented: Theorem 1. Let \(M\) be a Kenmotsu pseudo-Riemannian manifold. If \(g\) represents a \(\ast\)-\(\eta\)-Ricci soliton and the potential vector field \(V\) is an infinitesimal contact transformation, then \(V\) is strict and \((M, g)\) is Einstein. Theorem 2. Let \(M\) be a Kenmotsu pseudo-Riemannian manifold of dimension \(2n + 1\). If the metric \(g\) represents an almost \(\ast\)-\(\eta\)-Ricci gradient soliton, then the manifold \(M\) is \(\eta\)-Einstein. Moreover, if \(\xi\) leaves the scalar curvature \(s\) invariant, then \((M, g)\) is Einstein. Theorem 3. If a Kenmotsu pseudo-Riemannian manifold \(M\) admits an almost \(\ast\)-\(\eta\)-Ricci soliton with non-zero potential vector field \(V\) collinear to \(\xi\), then the manifold \(M\) is \(\eta\)-Einstein. Moreover, if \(\xi\) leaves the scalar curvature \(s\) invariant, then \(M\) is an Einstein manifold. Theorem 4. If a Kenmotsu pseudo-Riemannian manifold \(M\) admits a non-trivial almost \(\ast\)-\(\eta\)-Ricci soliton with \(V = F\xi\) for some constant \(F\), then this manifold is Einstein.
scientific article; zbMATH DE number 7612258
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