先进流形-度量对
Advanced manifold-metric pairs
AI总结：
本文提出一种新的数学形式化方法，用于构建高级流形-度量对，结合数学物理、场论、拓扑学、代数、概率和统计学概念，通过严谨的数学构造证明和逻辑基础，发展出广义流形-度量对，包括各向同性膨胀流形及其概率和熵变种，探讨了度量的高阶张量度量和复数、四元数陪域的应用。
摘要：
本文提出了一种新的数学形式化方法，用于高级流形-度量对，增强了几何和拓扑框架。我们使用函数方法构造各种D维流形及其相关度量空间，重点整合数学物理、场论、拓扑学、代数、概率和统计学的概念。我们的方法采用严谨的数学构造证明和逻辑基础，发展出广义流形-度量对，包括各向同性膨胀流形以及概率和熵变种。关键结果包括通过Urysohn度量定理建立拓扑流形的度量性，高阶张量度量的公式化，以及复数和四元数陪域的探索，应用于如膨胀时空模型等宇宙学模型。通过将时空广义集与信息论和概率方法结合，我们实现了统一框架，推进了对流形-度量相互作用及其物理意义的理解。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
