Advanced Manifold–Metric Pairs

Pierros Ntelis ()
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Pierros Ntelis: Independent Researcher, 13 Allee Turcat Mery, 13008 Marseille, France

Mathematics, 2025, vol. 13, issue 15, 1-29

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.

Keywords: mathematics; mathematical physics; manifolds; metrics; algebra; topology; probabilities; statistics; cosmology (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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