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Generalized quasi-Einstein manifolds with harmonic Weyl tensor. (English) Zbl 1246.53040

Summary: We introduce the notion of generalized quasi-Einstein manifold that generalizes the concepts of Ricci soliton, Ricci almost soliton and quasi-Einstein manifolds. We prove that a complete generalized quasi-Einstein manifold with harmonic Weyl tensor and with zero radial Weyl curvature is locally a warped product with \((n - 1)\)-dimensional Einstein fibers. In particular, this implies a local characterization for locally conformally flat gradient Ricci almost solitons, similar to that proved for gradient Ricci solitons.

MSC:

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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References:

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