A Note on Generalized Quasi-Einstein and ( λ, n + m )-Einstein Manifolds with Harmonic Conformal Tensor

Sameh Shenawy, Carlo Alberto Mantica, Luca Guido Molinari and Nasser Bin Turki
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Sameh Shenawy: Basic Science Department, Modern Academy for Engineering and Technology, Maadi, Egypt
Carlo Alberto Mantica: I.I.S. Lagrange, Via L. Modignani 65, 20161 Milan, Italy
Luca Guido Molinari: Physics Department, Università Degli Studi di Milano, Via Celoria 16, 20133 Milano, Italy
Nasser Bin Turki: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Mathematics, 2022, vol. 10, issue 10, 1-11

Abstract: Sufficient conditions for a Lorentzian generalized quasi-Einstein manifold M , g , f , μ to be a generalized Robertson–Walker spacetime with Einstein fibers are derived. The Ricci tensor in this case gains the perfect fluid form. Likewise, it is proven that a λ , n + m -Einstein manifold M , g , w having harmonic Weyl tensor, ∇ j w ∇ m w C j k l m = 0 and ∇ l w ∇ l w < 0 reduces to a perfect fluid generalized Robertson–Walker spacetime with Einstein fibers. Finally, M , g , w reduces to a perfect fluid manifold if φ = − m ∇ ln w is a φ R i c -vector field on M and to an Einstein manifold if ψ = ∇ w is a ψ R i c -vector field on M . Some consequences of these results are considered.

Keywords: ( ? , n + m )-Einstein manifolds; generalized quasi-Einstein manifold; perfect fluid; torse-forming vector fields (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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