Advanced manifold-metric pairs

TL;DR

This paper introduces a new mathematical framework for advanced manifold-metric pairs, integrating concepts from physics, topology, and probability to model complex geometric and cosmological structures.

Contribution

It develops generalized manifold-metric pairs with rigorous proofs, including probabilistic and tensor variants, advancing the mathematical modeling of spacetime and related physical theories.

Findings

Established metrizability of topological manifolds using Urysohn's theorem

Formulated higher-rank tensor metrics for complex geometries

Applied the framework to cosmological models like expanding spacetime

Abstract

This article presents a novel mathematical formalism for advanced manifold--metric pairs, enhancing the frameworks of geometry and topology. We construct various D-dimensional manifolds and their associated metric spaces using functional methods, with a focus on integrating concepts from mathematical physics, field theory, topology, algebra, probability, and statistics. Our methodology employs rigorous mathematical construction proofs and logical foundations to develop generalized manifold--metric pairs, including homogeneous and isotropic expanding manifolds, as well as probabilistic and entropic variants. Key results include the establishment of metrizability for topological manifolds via the Urysohn Metrization Theorem, the formulation of higher-rank tensor metrics, and the exploration of complex and quaternionic codomains with applications to cosmological models like the expanding spacetime. By combining spacetime generalized sets with information-theoretic and probabilistic approaches, we achieve a unified framework that advances the understanding of manifold--metric interactions and their physical implications.