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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Sameh Shenawy & Carlo Alberto Mantica & Luca Guido Molinari & Nasser Bin Turki, 2022. "A Note on Generalized Quasi-Einstein and ( λ , n + m )-Einstein Manifolds with Harmonic Conformal Tensor," Mathematics, MDPI, vol. 10(10), pages 1-11, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:10:p:1731-:d:818621
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