smooth manifold（光滑流形）

前置

• 子空间

定义

• A topological space $M$ is called an $m$-dimensional manifold if $M$ satisfies the following:

• Hausdorff: $M$ is a Hausdorff space

• Second countable: $M$ is a second countable space

• Locally Euclidean: For any $x\in M$, there is an open neighborhood $U_x$ of $x$ that is homeomorphic to some open set in ${\mathbb{R}}^m$.

• We call $(U,\varphi )$ a coordinate chart, or chart, if $U$ is an open set of $M$, $U^{\prime }$ is an open set of ${\mathbb{R}}^m$, and $\varphi$ is a homeomorphism from $U$ to $U^{\prime }$. (Note that homeomorphisms are continuous mappings, and continuous mappings are defined on topological spaces, so $U$ and $U^{\prime }$ here should be considered relative topological spaces.) $U$ is called the coordinate domain or coordinate neighborhood. If $\varphi (p)=0$, then the chart is said to be centered at p.
  If, in addition, $\varphi (U)$ is an open ball in ${\mathbb{R}}^n$, then $U$ is called a coordinate ball; if $\varphi (U)$ is an open cube, $U$ is a coordinate cube. The map $\varphi$ is called a (local) coordinate map, and the component functions $x_1,\dots ,x_n$ of $\varphi$, defined by $\varphi (p)=(x_1(p),\dots ,x_n(p))$, are called local coordinates on $U$. We sometimes write things such as “$(U,\varphi )$ is a chart containing $p$” as shorthand for “$(U,\varphi )$ is a chart whose domain $U$ contains $p$.” If we wish to emphasize the coordinate functions $x_1,\dots ,x_n$ instead of the coordinate map $\varphi$, we sometimes denote the chart by $(U,(x_1,\dots ,x_n))$ or $(U,(x_i))$.

性质

1. Every topological manifold has a countable basis of precompact coordinate balls.