Warped Product Manifolds in Relativity, Cosmology, and Astrophysics: From Geometric Foundations to Physical Applications

1. Introduction and RationaleWarped product manifolds represent one of the most profound bridges between differential geometry and theoretical physics, establishing a rigorous geometric framework that has fundamentally shaped our understanding of spacetime. Since their systematic introduction by Bishop and O'Neill in their seminal work on manifolds of negative curvature, warped products have proven indispensable for modeling the curved geometries that characterize general relativity. Their mathematical significance stems from a remarkable property: while standard Riemannian products of manifolds cannot admit negative curvature, warped products naturally accommodate such geometries, making them uniquely suited for describing the rich curvature structures of relativistic spacetimes.The occurrence of warped product structures in modern physics seems to be unavoidable. The Schwarzschild spacetime, Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological models, Kerr rotating black holes, etc., can be described by warped product geometries. On the one hand, it is of interest to study warped product manifolds with specific types of curvature. On the other hand, a physical interpretation of the obtained results can reveal new effects that can be observed in nature.Recently, there has been increasing interest in studying sequential doubly warped product (SD WP) manifolds. The SDWP structures make it possible to describe more complex physical situations, considering several hierarchically ordered warping functions. The study of curvature properties of such manifolds and of warped product manifolds in general is of great importance since curvature is a measure of gravitational field strength.The study of quasi-conformal curvature tensors, pseudo-projective curvature tensors, and W_2-curvature tensors on warped product manifolds has made it possible to relate them to dark energy, branes, geometrically induced dark matter halos, and other accompanying effects characteristic of these theories.The purpose of this Special Issue is to facilitate a meeting of mathematicians who study different aspects of warped product manifolds, including physicists interested in the properties of these manifolds. The results presented in this volume will provide an up-to-date overview of the study of these important mathematical objects. The collected papers will highlight the strengths of the warped product approach, both in terms of mathematical physics and mathematical problems. It will present works devoted to the study of various tensor curvature theories on warped product manifolds and their impact on cosmology and physics in general. 2. Scope and TopicsThe Special Issue will encompass, but is not limited to, the following thematic areas:A. Geometric Foundations Curvature tensors on warped, doubly warped, and sequential doubly warped product manifolds (Riemann, Ricci, conformal, projective, quasi-conformal, pseudo-projective, and W_2-tensors). Ricci solitons, gradient Ricci solitons, and Ricci-Bourguignon solitons on warped product structures. Einstein manifolds, quasi-Einstein structures, and constant curvature characterizations. Warped product submanifolds, slant curves, and their geometric invariants. Tensor analysis, conformal geometry, and submersions in warped product settings. Symmetry groups and Killing vector fields on warped product spacetimes.B. General Relativity and Spacetime Models Warped product representations of exact solutions to Einstein's field equations. Generalized Robertson-Walker (GRW) spacetimes and their curvature properties. Black hole solutions in warped product frameworks (Schwarzschild, Kerr, GMGHS, and beyond). Perfect fluid spacetimes, energy conditions, and fluid dynamics in warped geometries. Petrov classification and spacetime symmetries in warped product settings. Singularity theorems and global structure of warped product spacetimes.C. Cosmology and Astrophysics Anisotropic cosmological models via SDWP manifolds. Emergent dark energy behaviors from geometric constraints and modified gravity. Brane-world scenarios and gravity localization in warped extra dimensions. Geometric dark matter halos and modified mass profiles from warped structures. Cosmic microwave background anomalies and light propagation in anisotropic backgrounds. Relativistic particle dynamics and gravitational wave propagation in warped spacetimes. Frame-dragging effects and rotating systems in twisted warped products. Observational signatures and constraints on warped product cosmological models. Topic editors: Meraj Khan, SHYAMAL KUMAR HHUI Source: https://www.frontiersin.org/research-topics/84998/warped-product-manifolds-in-relativity-cosmology-and-astrophysics-from-geometric-foundations-to-physical-applications