【manifold】（物理学家用的）流形是什么？

第一性原理:
对于物理学家来说，最重要的术语是黎曼度量、列维-奇维塔联络he Levi-Civita-connection和黎曼曲率张量，为此你真的不需要知道豪斯多夫、同胚、平滑度和所有其他抽象术语的真正含义。一个好的流形介绍应该使用球体、环面或圆柱体等具体的例子，并解释这些具体例子的术语。The most important terms for physicists are the Riemann metric, the Levi-Civita-connection and the Riemann curvature tensor and for that you don't really need to know what Hausdorff, homeomorphisms, smoothness and all the other abstract terms really mean. A good introduction of manifolds should use concrete examples like a sphere, a torus or a cylinder and explain those terms for these specific examples.

第一性原理:
The mathematical definition of a manifold: A subset of R^n locally diffeomorphic at all points to a neighbourhood of the origin in R^m, where m<n. The engineering definition: Something that bends.