标题： 黎曼几何：定义、图像和结果 显示英文标题

标题： Riemannian Geometry: Definitions, Pictures, and Results

摘要： 提供了一个教学但简洁的黎曼几何概述，适用于物理学中的使用。 重点在于定义和可视化概念及其之间的关系，以及列出常见的混淆点、替代符号和行话，以及相关的事实和定理。 特别注意详细的图表和几何观点，其中一些似乎对文献来说是新颖的。 避免了教科书中已有详细内容的主题，如历史动机、证明和推导，以及用于实际计算的工具。 尽可能多地为带有连接的流形（省略度量）开发材料，以明确哪些方面可以轻易推广到规范理论。 在大多数情况下，不假设坐标框架或零扭转，并同时发展坐标无关、张量和卡当形式主义。 显示英文摘要

摘要： A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. As much material as possible is developed for manifolds with connection (omitting a metric) to make clear which aspects can be readily generalized to gauge theories. The presentation in most cases does not assume a coordinate frame or zero torsion, and the coordinate-free, tensor, and Cartan formalisms are developed in parallel.