Machine-translated from Chinese.
拓扑流形
第二可数性 + 局部紧 $\Rightarrow$ 仿紧性(任意开覆盖有开的,
局部有限的加细).
局部有限: 称覆盖$\{A_\alpha\}$是局部有限的, 若$\,\forall\,p\in M,$
$\,\exists\,$邻域$U\ni p$使得$U$只与有限个$A_\alpha$相交.
微分流形例子
研究微分流形上的微积分需要$C^\infty$微分结构$[\mathcal{F}=\{(U_\alpha,\varphi_\alpha)\}].$
$C^\infty$流形=拓扑流形+$C^\infty$结构.
$M$为$C^\infty$流形, $[\mathcal{F}]=[\{(U_\alpha,\varphi_\alpha)\}],$
$U\subset M$为开子集. 那么$U$也是拓扑流形,
同时有诱导的$C^\infty$坐标图册$[\mathcal{F}_U]=[\{U_\alpha\cap U,\varphi_\alpha|_{U\cap U_\alpha}\}]$,
从而$U$也是$C^\infty$流形, 称为开子流形.
$m$阶方阵全体$M_m=\{(a_{ij})_{m\times m}|a_{ij}\in \mathbb{R}\}\leftrightarrow \mathbb{R}^{m^2},$
为向量空间的同构. 因此也可视为一个$C^\infty$流形.
$GL(m,\mathbb{R}):=\{A\in M_{m}|\det A\neq 0\}\subset M_m=\mathbb{R}^{m^2}.$
$\det:M_m\rightarrow \mathbb{R}$为连续函数.
由于$GL(m,\mathbb{R})=(\det)^{-1}(\mathbb{R}\setminus \{0\})$为$M_m$的开子集,
它是开子流形. 因此$GL(m,\mathbb{R})$也是一个$m^2$维$C^\infty$流形,
称为($\mathbb{R}$上)一般线性群, 以矩阵乘法为群上运算.
后面将说明, 矩阵乘法与求逆运算均为$C^\infty$流形间的$C^\infty$映照,
给出了其上两种结构($C^\infty$流形结构与群结构)之间的关系.
这样的群称为李群(Lie Group).
设$(M^m,[\mathcal{F}=\{(U_{\alpha},\varphi_\alpha)\}]),$
$(N^n,[\widetilde {\mathcal{F} }=\{(V_\beta,\psi_\beta)\}])$是$C^\infty$流形.
赋予$M^m\times N^n$积拓扑,
即$U\times V$为开子集当且仅当两者分别为各自空间开子集.
容易验证$M^m\times N^n$是一个$m+n$维拓扑流形,
以$[\mathcal{F}\times \widetilde {\mathcal{F} }:=\{(U_{\alpha}\times V_{\beta},\varphi_\alpha\times \psi_\beta)\}]$为$C^\infty$坐标图册,
成为$C^\infty$微分流形. 称$M\times N$为$M,N$的积流形.
可归纳定义$M_1\times\cdots\times M_k.$
$\mathbb{R}^{m+1}\setminus \{0\}$引入等价关系$\sim,$
$x\sim y\Leftrightarrow x=\lambda y.$
商空间$M=\mathbb{R}^{m+1}\setminus\{0\}/_\sim,$ 赋予商拓扑,
即$U\subset M$为开子集$\Leftrightarrow$ $\pi^{-1}(U)$为原空间开子集,
其中$\pi:x\mapsto [x]$为自然投影. $[x]$可视为过原点的直线,
或球面上的两点粘贴. 可以说明商拓扑是Hausdorff, 第二可数的.
令开集$U_i=\{[x=(x^1,\cdots,x^{m+1})]\in M|x^i\neq 0\},$
$(x^1,\cdots,x^{m+1})$称为$[x]$的齐次坐标.
$M=\bigcup_{i=1}^{m+1} U_i.$
定义同胚$\varphi_i:U_i\rightarrow \mathbb{R}^m,$
$[x]\mapsto\frac{1}{x_i}(x^1,\cdots, \widehat{x^{i} },\cdots, x^{m+1}).$
$(U_i,\varphi_i)$即为$M$的一个坐标图册. 称对应的坐标为非齐次坐标.
记$\xi_{(i)}^k=\frac{x^k}{x^i}.$ 任取$1\le i<j\le m+1,$
$U_i\cap U_j$上坐标变换为$\xi_{(i)}^j\mapsto \xi_{(j)}^i=\frac{1}{\xi_{(i)}^j},$
$\xi_{(i)}^l\mapsto \xi_{(j)}^l=\frac{\xi^l_{(i)} }{\xi_{(i)}^j},$
$l\neq i,j,$ 是$\mathbb{R}^m$上的$C^\infty$映射,
因此坐标图册是$C^\infty$的.
从而$(M,[\{(U_i,\varphi_i)\}])$成为$m$维$C^\infty$流形,
记为实射影空间$\mathbb{R}\mathrm{P}^m.$
取$\mathbb{R}^N$为$N$维欧氏空间,
记$G_{k,N}=\{\text{$P$为$\mathbb{R}^N$的$k$维子空间}\}.$
赋予拓扑与$C^\infty$结构使之成为一个$C^\infty$流形, 称为Grassman流形.
易见$G_{1,N}=\mathbb{R}\mathrm{P}^{N-1}.$
流形间的映射
设$(M^m,[\mathcal{F}])$是$C^\infty$流形, $f:M\rightarrow \mathbb{R}.$
$\,\forall\,p\in M,$
若存在含$p$的允许坐标系$(U,\varphi)$使$f\circ \varphi^{-1}\in C^\infty(\varphi(U)),$
则称$f$在$p$点是$C^\infty$的.
称$(f\circ \varphi^{-1})(x^1,\cdots,x^m)$为局部表示.
这里允许坐标系指包含在$[\mathcal{F}]$极大坐标图册中的坐标卡.
如果$f$点点$C^\infty,$ 则称$f\in C^\infty(M).$
注 1.1. 上述概念与坐标系选取无关. 注意$f$在$p$点$C^\infty$等价于$f\circ \varphi^{-1}$在$\varphi(p)$某个小邻域$C^\infty.$
注 1.2. 类似可给出$f$在$p$点是$C^r$的, 以及$f\in C^r(M)$的定义.
文章最后更新于 2021-09-25 16:22:50
topological manifold
Local road connectivity (coordinate neighborhood);
Connected $\Leftrightarrow$ road connected (local road connected);
Connected components are countable (second countability);
Locally compact (locally Euclidean).
Second countability + local compactness $\Rightarrow$ Imitation tightness (Any opening covers those that are open,
Limited local detailing).
Locally finite: The coverage $\{A_\alpha\}$ is said to be locally limited , if$\,\forall\,p\in M,$
$\,\exists\,$ neighborhood $U\ni p$ makes $U$ intersect only a limited number of $A_\alpha$.
Differential manifold example
Studying calculus on differential manifolds requires $C^\infty$ Differential Structures $[\mathcal{F}=\{(U_\alpha,\varphi_\alpha)\}].$
$C^\infty$ manifold = topological manifold + $C^\infty$ structure.
$M$ is the $C^\infty$ manifold, $[\mathcal{F}]=[\{(U_\alpha,\varphi_\alpha)\}],$
$U\subset M$ is an open subset. Then $U$ is also a topological manifold,
At the same time, there is an induced $C^\infty$ coordinate atlas $[\mathcal{F}_U]=[\{U_\alpha\cap U,\varphi_\alpha|_{U\cap U_\alpha}\}]$,
Therefore $U$ is also a $C^\infty$ manifold, which is called open submanifold.
$m$ whole square array $M_m=\{(a_{ij})_{m\times m}|a_{ij}\in \mathbb{R}\}\leftrightarrow \mathbb{R}^{m^2},$
is the isomorphism of vector space. Therefore it can also be regarded as a $C^\infty$ manifold.
$GL(m,\mathbb{R}):=\{A\in M_{m}|\det A\neq 0\}\subset M_m=\mathbb{R}^{m^2}.$
$\det:M_m\rightarrow \mathbb{R}$ is a continuous function.
Since $GL(m,\mathbb{R})=(\det)^{-1}(\mathbb{R}\setminus \{0\})$ is an open subset of $M_m$,
It is an open submanifold. Therefore $GL(m,\mathbb{R})$ is also a $m^2$-dimensional $C^\infty$ manifold,
Called ($\mathbb{R}$ on) general linear group, using matrix multiplication as an operation on the group.
It will be explained later that matrix multiplication and inversion operations are $C^\infty$ mappings between $C^\infty$ manifolds.
The relationship between the above two structures ($C^\infty$ manifold structure and group structure) is given.
Such a group is called Lie Group.
Let $(M^m,[\mathcal{F}=\{(U_{\alpha},\varphi_\alpha)\}]),$
$(N^n,[\widetilde {\mathcal{F} }=\{(V_\beta,\psi_\beta)\}])$ is the $C^\infty$ manifold.
Given the $M^m\times N^n$ product topology,
That is, $U\times V$ is an open subset if and only if both are open subsets of their respective spaces.
It is easy to verify that $M^m\times N^n$ is a $m+n$-dimensional topological manifold,
Take $[\mathcal{F}\times \widetilde {\mathcal{F} }:=\{(U_{\alpha}\times V_{\beta},\varphi_\alpha\times \psi_\beta)\}]$ as $C^\infty$ coordinate atlas,
Becomes $C^\infty$ differential manifold. Call $M\times N$ $M,N$ Accumulative manifold.
Inducible definition $M_1\times\cdots\times M_k.$
$\mathbb{R}^{m+1}\setminus \{0\}$ Introduce equivalence relations $\sim,$
$x\sim y\Leftrightarrow x=\lambda y.$
Quotient space $M=\mathbb{R}^{m+1}\setminus\{0\}/_\sim,$ given quotient topology,
That is, $U\subset M$ is an open subset $\Leftrightarrow$ and $\pi^{-1}(U)$ is an open subset of the original space.
Where $\pi:x\mapsto [x]$ is natural projection. $[x]$ can be regarded as a straight line passing through the origin,
Or two points on the sphere are pasted. It can be explained The quotient topology is Hausdorff, the second countable.
Let the open set $U_i=\{[x=(x^1,\cdots,x^{m+1})]\in M|x^i\neq 0\},$
$(x^1,\cdots,x^{m+1})$ is called $[x]$ Homogeneous coordinates.
$M=\bigcup_{i=1}^{m+1} U_i.$
Define homeomorphism$\varphi_i:U_i\rightarrow \mathbb{R}^m,$
$[x]\mapsto\frac{1}{x_i}(x^1,\cdots, \widehat{x^{i} },\cdots, x^{m+1}).$
$(U_i,\varphi_i)$ is a coordinate atlas of $M$. The corresponding coordinates are called non-homogeneous coordinates.
Remember $\xi_{(i)}^k=\frac{x^k}{x^i}.$, take any $1\le i<j\le m+1,$
The upper coordinates of $U_i\cap U_j$ are transformed to $\xi_{(i)}^j\mapsto \xi_{(j)}^i=\frac{1}{\xi_{(i)}^j},$
$\xi_{(i)}^l\mapsto \xi_{(j)}^l=\frac{\xi^l_{(i)} }{\xi_{(i)}^j},$
$l\neq i,j,$ is the mapping of $C^\infty$ on $\mathbb{R}^m$,
Therefore, the coordinate atlas is $C^\infty$.
Thus $(M,[\{(U_i,\varphi_i)\}])$ becomes a $m$-dimensional $C^\infty$ manifold,
recorded as real projective space $\mathbb{R}\mathrm{P}^m.$
Take $\mathbb{R}^N$ as $N$-dimensional Euclidean space,
Note$G_{k,N}=\{\text{$P$为$\mathbb{R}^N$的$k$维子空间}\}.$
Endow the topology and $C^\infty$ structure to make it a $C^\infty$ manifold, called a Grassman manifold.
Easy to see$G_{1,N}=\mathbb{R}\mathrm{P}^{N-1}.$
Mapping between manifolds
Let $(M^m,[\mathcal{F}])$ be the $C^\infty$ manifold, $f:M\rightarrow \mathbb{R}.$
$\,\forall\,p\in M,$
If there is $p$ Allow coordinate systems $(U,\varphi)$ enable $f\circ \varphi^{-1}\in C^\infty(\varphi(U)),$
Then $f$ is said to be $C^\infty$ at point $p$.
Call $(f\circ \varphi^{-1})(x^1,\cdots,x^m)$ local representation.
here Allow coordinate systems Refers to the coordinate card included in the $[\mathcal{F}]$ maximum coordinate atlas.
If $f$ points to $C^\infty,$, it is called $f\in C^\infty(M).$
Note 1.1. The above concept has nothing to do with the selection of the coordinate system. Note that $f$ at $p$ point $C^\infty$ is equivalent to $f\circ \varphi^{-1}$ at $\varphi(p)$ in a small neighborhood $C^\infty.$
Note 1.2. Similarly, $f$ is $C^r$ at point $p$, and the definition of $f\in C^r(M)$ can be given.
The article was last updated on 2021-09-25 16:22:50
