Geometry of Ricci-Yamabe solitons on four-dimensional Lorentzian manifolds

Abdul Haseeb; Sudhakar Kumar Chaubey; Aydin Gezer; Mohammad Nazrul Islam Khan; Sibel Turanli; Abdul Haseeb; Sudhakar Kumar Chaubey; Aydin Gezer; Mohammad Nazrul Islam Khan; Sibel Turanli

doi:10.3934/era.2026367

Electronic Research Archive

2026, Volume 34, Issue 12: 8719-8739. doi: 10.3934/era.2026367

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Research article

Geometry of Ricci-Yamabe solitons on four-dimensional Lorentzian manifolds

1.
Department of Mathematics, College of Science, Jazan University, Jazan 45142, Saudi Arabia
2.
Unit of Sciences and Mathematics, Department of Supportive Requirements, University of Technology and Applied Sciences, Shinas 324, Oman
3.
Department of Mathematics, Faculty of Science, Ataturk University, Erzurum 25240, Turkiye
4.
Department of Computer Engineering, College of Computer, Qassim University, Buraydah 51452, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum 25050, Turkiye

Received: 28 July 2026 Revised: 21 September 2026 Accepted: 24 September 2026 Published: 22 October 2026

We investigate Ricci-Yamabe solitons (RYSs) and gradient Ricci-Yamabe solitons (GRYSs) on four-dimensional Lorentzian manifolds $ (M^4, g) $ admitting a unit timelike torse-forming vector field. We show that when $ V = \zeta $ and $ \rho\neq0 $, the soliton equation is compatible with a perfect fluid spacetime structure. For the case of a constant scalar curvature, it is shown that if $ (M^4, g) $ admits an RYS or a GRYS, then $ (M^4, g) $ is an Einstein spacetime, the associated soliton vector field is Killing, and consequently, the RYS or GRYS is trivial. In the non-constant scalar curvature case, the scalar curvature is shown to satisfy a screened wave (Klein-Gordon-type) equation in the RYS setting, whereas the potential function of a GRYS satisfies a second-order differential equation governing its convexity and concavity along the integral curves of the unit timelike vector field. Finally, a specific constant-curvature example is presented in which the GRYS becomes trivial under the corresponding parameter relations.
- Ricci-Yamabe solitons,
- Einstein spacetimes,
- Lorentzian manifolds,
- screened wave equation,
- partial differential equations
Citation: Abdul Haseeb, Sudhakar Kumar Chaubey, Aydin Gezer, Mohammad Nazrul Islam Khan, Sibel Turanli. Geometry of Ricci-Yamabe solitons on four-dimensional Lorentzian manifolds[J]. Electronic Research Archive, 2026, 34(12): 8719-8739. doi: 10.3934/era.2026367

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Abstract

We investigate Ricci-Yamabe solitons (RYSs) and gradient Ricci-Yamabe solitons (GRYSs) on four-dimensional Lorentzian manifolds $ (M^4, g) $ admitting a unit timelike torse-forming vector field. We show that when $ V = \zeta $ and $ \rho\neq0 $, the soliton equation is compatible with a perfect fluid spacetime structure. For the case of a constant scalar curvature, it is shown that if $ (M^4, g) $ admits an RYS or a GRYS, then $ (M^4, g) $ is an Einstein spacetime, the associated soliton vector field is Killing, and consequently, the RYS or GRYS is trivial. In the non-constant scalar curvature case, the scalar curvature is shown to satisfy a screened wave (Klein-Gordon-type) equation in the RYS setting, whereas the potential function of a GRYS satisfies a second-order differential equation governing its convexity and concavity along the integral curves of the unit timelike vector field. Finally, a specific constant-curvature example is presented in which the GRYS becomes trivial under the corresponding parameter relations.

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