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

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Abstract

We investigate Ricci-Yamabe solitons (RYSs) and gradient Ricci-Yamabe solitons (GRYSs) on four-dimensional Lorentzian manifolds (M⁴,g) admitting a unit timelike torse-forming vector field. We show that when V = ζ and ρ , 0, the soliton equation is compatible with a perfect fluid spacetime structure. For the case of a constant scalar curvature, it is shown that if (M⁴,g) admits an RYS or a GRYS, then (M⁴,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.

Keywords

Ricci-Yamabe solitons / Einstein spacetimes / Lorentzian manifolds / screened wave equation / partial differential equations

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