Hyperbolic Ricci Solitons on Sequential Doubly Warped Product Manifolds: Existence, Classification, and Applications

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.