Quasi-conformal curvature tensor on sequential doubly warped product manifolds with applications

Abstract This paper introduces a comprehensive geometric framework that bridges two important but previously disconnected areas of Riemannian geometry: the theory of sequential doubly warped product (SDWP) manifolds and the theory of quasi-conformal curvature tensors. We present the first systematic investigation of quasi-conformal curvature properties in the context of SDWP manifolds – a hierarchical geometric structure that substantially generalizes classical warped products and offers enhanced modeling capabilities for multi-scale physical theories. Our main contribution lies in establishing complete characterizations of SDWP manifolds satisfying three fundamental curvature conditions: quasi-conformal flatness, quasi-conformal symmetry, and divergence-free quasi-conformal curvature. The results reveal that these geometric constraints impose remarkably severe restrictions on both the factor manifolds and the warping functions, leading to new hierarchical PDE systems that govern the warping functions. Notably, we demonstrate that quasi-conformal flatness forces the base manifolds to be Einstein or of constant curvature, while simultaneously requiring the warping functions to satisfy specific coupled partial differential equations derived from the global curvature structure. The physical significance of our work is substantiated through several innovative applications in modern gravity and cosmology, including: (i) anisotropic cosmological models exhibiting emergent dark energy behavior, (ii) brane-world scenarios with geometrically constrained gravity localization, and (iii) novel black hole solutions surrounded by geometric dark matter halos. These applications demonstrate how our derived geometric constraints naturally emerge in physically relevant settings, thereby establishing a concrete bridge between abstract differential geometry and realistic spacetime models.