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Ricci-like solitons on almost contact B-metric manifolds. (English) Zbl 1442.53034

In the paper under review an almost contact B-metric manifold is a \((2n+1)\)-dimensional manifold endowed with an almost contact structure \((\varphi , \xi , \eta )\) and a semi-Riemannian compatible metric \(g\) of signature \((n+1, n)\). This gives a new semi-Riemannian metric \(\tilde{g}(\cdot , \cdot )=g(\cdot , \varphi \cdot )+\eta \otimes \eta (\cdot , \cdot )\) also of signature \((n+1, n)\). The initial pair \((g, \xi )\) is called Ricci-like soliton if there exist three constants \(\lambda , \mu , \nu \) such that \(\frac{1}{2}\mathcal{L}_{\xi }g+\rho +\lambda g+\mu \tilde{g}+\nu \eta \otimes \eta =0\) with \(\rho \) the Ricci tensor field of \(g\). Hence, this work deals with four subjects: Sasaki-like B-metrics, Einstein-like B-metrics, torse-forming \(\xi \) and Ricci-like solitons. There are some interesting examples.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53D15 Almost contact and almost symplectic manifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53D35 Global theory of symplectic and contact manifolds
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
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References:

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