基于度量张量Takagi分解的Riemannian Geometry
Riemannian Geometry Based on the Takagi's Factorization of the Metric Tensor
摘要：
黎曼几何是现代数学和物理学的主要理论内容之一。有关文献中对黎曼几何的研究都是利用一条定义明确的解析路径来进行的，通常是从度量的概念作为基本概念出发，以联络作为中间几何对象来研究曲率及其性质。本文提出了一种分析黎曼几何的新途径。它是基于一组中间的几何对象从高木因式分解的度量张量。这些中间对象为几何分析提供了一个新的视角，为弯曲与平坦流形提供了条件，并且还提供了曲率张量在规范部分的新分解，这对理论物理学很有用。
英文摘要：
The Riemannian geometry is one of the main theoretical pieces in Modern Mathematics and Physics. The study of Riemann Geometry in the relevant literature is performed by using a well defined analytical path. Usually it starts from the concept of metric as the primary concept and by using the connections as an intermediate geometric object, it is achieved the curvature and its properties. This paper presents a different analytical path to analyze the Riemannian geometry. It is based on a set of intermediate geometric objects obtained from the Takagi's factorization of the metric tensor. These intermediate objects allow a new viewpoint for the analysis of the geometry, provide conditions for the curved vs. flat manifolds, and also provide a new decomposition of the curvature tensor in canonical parts, which can be useful for Theoretical Physics.
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