Ayman Elsharkawy

Twisted spacetimes generalize Robertson–Walker cosmological models by allowing the scale factor to depend on both spatial position and time, thereby modeling inhomogeneous cosmic expansion. This paper establishes geometric conditions under which a Lorentzian manifold becomes a twisted spacetime and determines when such spacetimes reduce to homogeneous generalized Robertson–Walker (GRW) models. We prove that Lorentzian manifolds admitting semi-symmetric metric connections with covariantly constant torsion tensor possess twisted spacetime structure. A spacetime with quasi-constant curvature becomes twisted under a natural condition on the gradients of its defining curvature scalars. We demonstrate that a twisted spacetime reduces to a GRW spacetime if and only if the difference between the two scalar functions occurring in its Ricci decomposition is an eigenvalue of the Ricci tensor with respect to the generating time-like vector, and that Ricci-symmetric twisted spacetimes are necessarily GRW. Conformally flat twisted spacetimes are shown to be a special subclass of generalized quasi-Einstein manifolds or a perfect fluid spacetime, reducing to GRW under specific conditions, including Ricci recurrence. These results identify geometric constraints distinguishing inhomogeneous from homogeneous cosmological models. Finally, we construct some physically relevant cosmological metric of twisted form that verifies our results.

This paper presents a comprehensive study of hyperbolic Ricci solitons on sequential doubly warped product manifolds (SDWPM). We establish a complete geometric framework for SDWPM, deriving explicit formulas for the Levi‐Civita connection, Riemann curvature tensor, Ricci tensor, and scalar curvature. Our main results include necessary and sufficient conditions for the existence of both gradient and non‐gradient hyperbolic Ricci solitons on these manifolds, complete classification theorems, and rigidity results for compact cases. We prove several novel existence theorems and provide explicit constructions of hyperbolic Ricci solitons on various SDWPM configurations. The relationship between hyperbolic Ricci solitons and Einstein metrics on SDWPM is thoroughly investigated, revealing deep connections between soliton structures and the underlying Riemannian geometry. Applications to cosmological models and geometric evolution equations demonstrate the physical relevance of our theoretical framework. The work significantly extends previous results on warped product manifolds and provides powerful new tools for studying geometric flows in complex geometric settings.

This paper introduces twisted sequential warped products. This new class of warped products extends solutions to Einstein’s field equations. We derive explicit formulas for covariant derivatives, curvature tensors, Ricci curvature, and scalar curvature. We characterize Einstein conditions and analyze geodesics. We also study conformal vector fields, including Killing and concircular types. Furthermore, we examine gradient almost Ricci solitons and the concircular curvature tensor. Explicit examples of such manifolds are presented, and applications to cosmology, black hole physics, and quantum field theory are discussed. These findings deepen the understanding of warped product geometry in general relativity. They also provide a foundation for further research.

We investigate Ricci–Bourguignon solitons on mixed doubly sequential warped product manifolds. Necessary and sufficient conditions for the existence of such solitons are established, and their implications for Einstein manifolds are analyzed. This work extends previous results on warped product manifolds to the more complex case of mixed doubly sequential warped products. We derive explicit formulas for Ricci tensor components, establish conditions for Einstein metrics, and study gradient solitons and conformal vector fields. Our classification theorems reveal fundamental separability constraints on the warping functions. Applications to cosmological models and black hole solutions demonstrate the physical relevance of these results.