Machine-translated from Chinese.
流形基本概念与例子
拓扑
$M,N$为两拓扑空间, $\varphi:M\rightarrow N$为双射,
且$\varphi$和$\varphi^{-1}$都是连续的,
则称$\varphi:M\rightarrow N$为同胚. 同胚的拓扑空间视为一样的.
如果拓扑空间具有可数拓扑基, 则称$M$满足第二可数公理.
设$M$为拓扑空间,
如果$\,\forall\,p,q\in M$存在含$p$和$q$的开邻域$U$和$V$,
使$U\cap V=\varnothing$, 则称$M$是Hausdorff的(满足第二分离公理).
欧氏空间$\mathbb{R}^m$为向量空间, 其上有加法, 有数乘.
一般用$l^2$距离定义$\mathbb{R}^m$上距离$d,$
$(\mathbb{R}^m,d)$成为度量空间, 由开球$B(x,r)$全体定义拓扑(构成拓扑基).
给定开集$U\subset\mathbb{R}^m,$ 函数$f:U\rightarrow \mathbb{R}.$
若$0\le r\le \infty,$ $f$直到$r$阶偏导存在且连续, 则记$f\in C^r(U).$
若$f$可展开为幂级数, 称其为解析函数, 记$f\in C^\omega(U).$
设$F:U\rightarrow V,$ $U\subset \mathbb{R}^m,$
$V\subset \mathbb{R}^n$为开子集,
$F(x)=(f^1(x^1,\cdots x^m),\cdots, f^n(x^1,\cdots, x^m)).$
若$f^i\in C^r(U),$ $1\le i\le n,$ 则记$F\in C^r(U,V).$
例子
曲线- $F:(a,b)\rightarrow \mathbb{R}^3$
曲面片- 同胚$F:U\subset \mathbb{R}^2\rightarrow \mathbb{R}^3$
任意过点$p$曲线可表为$C(t)=F(u(t),v(t)),$
切向量$C'(t_0)=F_uu'(t_0)+F_vv'(t_0).$ 默认要求$F_u\times F_v\neq 0,$
称其满足正则性. 切向量全体张成一个平面, 称为切平面.
球面无法用单张曲面片来描述(紧性, 同调群...) 但可以用若干张曲面片来描述.
流形
设$M$为拓扑空间, 满足$\,\forall\,p\in M,$ 存在含$p$的邻域$U_p$,
及从$U_p$到$m$维欧氏空间$\mathbb{R}^m$中开集$D$的同胚$\varphi:U_p\rightarrow D.$
即它是局部欧氏的. 若其还满足第二可数公理与Hausdorff性,
则称$M$为$m$维拓扑流形. 流形的概念是曲线,曲面概念高维,内蕴的推广,
可视为一块块欧氏空间粘贴起来的一种拓扑空间.
设$M^m$为$m$维拓扑流形, 称$(U,\varphi)$为局部坐标系/图/卡,
$x^i(q)$称为$q$在$(U,\varphi)$中第$i$个坐标($1\le i\le m$).
若$(U,\varphi),(V,\psi)$为两个坐标系, $U\cap V\neq \varnothing,$
则$\psi\circ \varphi^{-1}$称为坐标变换.
如果$\varphi\circ \psi^{-1},\psi\circ \varphi^{-1}\in C^r,$
则称$(U,\varphi)$与$(V,\psi)$是$C^r$相容的.
它们当然至少是$C^0$相容的.
设$\mathcal{F}=\{(U_\alpha,\varphi_\alpha)\}_{\alpha}$是拓扑流形$M^m$的一族坐标图.
如果$\bigcup_{\alpha\in \Lambda} U_{\alpha}=M,$
则称$\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in \Lambda}$是$M$的坐标图册.
如$\,\forall\,\alpha,\beta\in \Lambda,$
$(U_\alpha,\varphi_\alpha),(U_\beta,\varphi_\beta)$是$C^r$相容的,
则记$\mathcal{F}\in C^r.$
分析流形上函数$f:M\rightarrow \mathbb{R}$时,
只需将$M$中的点在局部坐标系上考虑,
那么$f\circ\varphi^{-1}$即为欧氏空间到欧氏空间上的函数,
称为$f$的局部表示, 以分析$f$的性质.
易见两种局部表示可由坐标变换转化, 若坐标变换是光滑的,
那么局部表示的微分性质在变换下保持, 即不依赖坐标系选取.
可见为了在流形上研究微分学, 必须要有好的坐标图册.
设$\mathcal{F}_1=\{(U_\alpha,\varphi_\alpha)\},$
$\mathcal{F}_2=\{(V_\beta,\psi_\beta)\}$是$M$上两个$C^\infty$坐标图册,
若$\,\forall\,\alpha,\beta,$
$(U_\alpha,\varphi_\alpha)$与$(V_\beta,\psi_\beta)$是$C^\infty$相容的,
则称$\mathcal{F}_1$与$\mathcal{F}_2$是$C^\infty$等价的,
记为$\mathcal{F}_1\sim \mathcal{F}_2.$
$M$上所有$C^\infty$坐标图册可划分为等价类集,
每一个$[\mathcal{F}]$称为$M$上一个$C^\infty$微分结构.
定义 1.1. $m$维$C^\infty$流形=$m$维拓扑流形+$C^\infty$结构\[$\mathcal{F}$\].
注 1.2. 类似地可以引入$C^r$微分结构, $C^\omega$微分结构, 定义$C^r$微分流形, $C^\omega$微分流形(解析流形).
注 1.3. 设$[\mathcal{F}]$是$M^m$上$C^\infty$结构, $\mathcal{F}_1=\{(U_\alpha,\varphi_\alpha)\}\in [\mathcal{F}].$ 若$(V,\psi)$与其中每个坐标图$C^\infty$相容, 可加入该图, 得到扩充的坐标图册$\mathcal{F}_1'.$ 它也是$C^\infty$坐标图册, 且仍在等价类$[\mathcal{F}]$中. 这样可得到极大$C^\infty$坐标图册$\mathcal{F}_{\max}\in [\mathcal{F}],$ 也称为$C^\infty$结构.
对$M=\mathbb{R}^m,$
取$\mathcal{F}=\{(\mathbb{R}^m,\mathrm{id})\}$即为$C^\infty$
$(C^\omega)$ 坐标图册,
称$[\mathcal{F}]$为$\mathbb{R}^m$上标准$C^\infty$结构,
$(\mathbb{R}^m,[\mathcal{F}]^{C^\infty})$为$m$维$C^\infty$流形.
文章最后更新于 2021-09-15 16:12:46
Basic concepts and examples of manifolds
Topology
$M,N$ is two topological spaces, $\varphi:M\rightarrow N$ is a bijection,
And $\varphi$ and $\varphi^{-1}$ are both continuous,
Then $\varphi:M\rightarrow N$ is called Homeomorphism. Homeomorphic topological spaces are considered the same.
If the topological space has countable topological bases, then $M$ is said to satisfy second countable axiom.
Let $M$ be the topological space,
If $\,\forall\,p,q\in M$ has open neighbors $U$ and $V$ containing $p$ and $q$,
If $U\cap V=\varnothing$ is used, then $M$ is said to be Hausdorff (satisfies the second separation axiom).
Euclidean space $\mathbb{R}^m$ is a vector space, on which there are additions and multiplications.
Generally, the distance $l^2$ is used to define the distance $d,$ on $\mathbb{R}^m$.
$(\mathbb{R}^m,d)$ becomes a metric space, and the topology is defined by the ball $B(x,r)$ as a whole (constituting the topological basis).
Given open set $U\subset\mathbb{R}^m,$ function $f:U\rightarrow \mathbb{R}.$
If $0\le r\le \infty,$ $f$ to $r$ order partial derivatives exist and are continuous, then record $f\in C^r(U).$
If $f$ can be expanded into a power series, it is called analytic function, remember $f\in C^\omega(U).$
Let $F:U\rightarrow V,$ $U\subset \mathbb{R}^m,$
$V\subset \mathbb{R}^n$ is an open subset,
$F(x)=(f^1(x^1,\cdots x^m),\cdots, f^n(x^1,\cdots, x^m)).$
If $f^i\in C^r(U),$ $1\le i\le n,$, record $F\in C^r(U,V).$
Example
Curve-$F:(a,b)\rightarrow \mathbb{R}^3$
Surface Patch-Homeomorphism$F:U\subset \mathbb{R}^2\rightarrow \mathbb{R}^3$
Any curve passing through point $p$ can be expressed as $C(t)=F(u(t),v(t)),$
Tangent vector $C'(t_0)=F_uu'(t_0)+F_vv'(t_0).$ Default requirement $F_u\times F_v\neq 0,$
call it satisfying Regularity. The tangent vectors all span into a plane, which is called tangent plane.
The spherical surface cannot be described by a single surface patch (compactness, homology group...) but can be described by several surface patches.
manifold
Assume $M$ is a topological space, satisfying $\,\forall\,p\in M,$ there is a neighborhood $U_p$ containing $p$,
And the homeomorphism $\varphi:U_p\rightarrow D.$ of the open set $D$ in the dimensional Euclidean space $\mathbb{R}^m$ from $U_p$ to $m$
i.e. it is local euclidean . If it also satisfies the second countability axiom and Hausdorff property,
Then $M$ is called $m$ dimension topological manifold. The concept of manifold is the concept of curve and surface High-dimensional, intrinsic promotion,
It can be regarded as a topological space in which pieces of Euclidean space are pasted together.
Let $M^m$ be the $m$-dimensional topological manifold, and call $(U,\varphi)$ Local coordinate system/map/card,
$x^i(q)$ is called the $i$th coordinate ($1\le i\le m$) of $q$ in $(U,\varphi)$.
If $(U,\varphi),(V,\psi)$ is two coordinate systems, $U\cap V\neq \varnothing,$
Then $\psi\circ \varphi^{-1}$ is called Coordinate transformation.
if$\varphi\circ \psi^{-1},\psi\circ \varphi^{-1}\in C^r,$
Then $(U,\varphi)$ and $(V,\psi)$ are said to be $C^r$ compatible of.
They are certainly at least $C^0$ compatible.
Let $\mathcal{F}=\{(U_\alpha,\varphi_\alpha)\}_{\alpha}$ be a family of coordinate graphs of the topological manifold $M^m$.
if$\bigcup_{\alpha\in \Lambda} U_{\alpha}=M,$
Then $\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in \Lambda}$ is said to be of $M$ Coordinate atlas.
Such as $\,\forall\,\alpha,\beta\in \Lambda,$
$(U_\alpha,\varphi_\alpha),(U_\beta,\varphi_\beta)$ is compatible with $C^r$,
Then record $\mathcal{F}\in C^r.$
When analyzing the function $f:M\rightarrow \mathbb{R}$ on the manifold,
Just consider the points in $M$ on the local coordinate system,
Then $f\circ\varphi^{-1}$ is the function from Euclidean space to Euclidean space,
called $f$ local representation, to analyze the properties of $f$.
It is easy to see that the two local representations can be transformed by coordinate transformation. If the coordinate transformation is smooth,
Then the differential properties of the local representation are maintained under transformation, i.e. Does not depend on coordinate system selection.
It can be seen that in order to study differential calculus on manifolds, a good coordinate atlas is necessary.
Let $\mathcal{F}_1=\{(U_\alpha,\varphi_\alpha)\},$
$\mathcal{F}_2=\{(V_\beta,\psi_\beta)\}$ is the two $C^\infty$ coordinate atlases above $M$,
If$\,\forall\,\alpha,\beta,$
$(U_\alpha,\varphi_\alpha)$ and $(V_\beta,\psi_\beta)$ are compatible with $C^\infty$,
Then $\mathcal{F}_1$ and $\mathcal{F}_2$ are called $C^\infty$ Equivalent of,
Recorded as $\mathcal{F}_1\sim \mathcal{F}_2.$
All $C^\infty$ coordinate atlases on $M$ can be divided into equivalent clusters,
Each $[\mathcal{F}]$ is called $M$ and the previous $C^\infty$ differential structure.
Definition 1.1. $m$-dimensional $C^\infty$ manifold = $m$-dimensional topological manifold + $C^\infty$ structure \[$\mathcal{F}$\].
Note 1.2. Similarly, $C^r$ differential structure, $C^\omega$ differential structure can be introduced, and $C^r$ differential manifold and $C^\omega$ differential manifold (analytical manifold) can be defined.
Note 1.3. Assume $[\mathcal{F}]$ is the $C^\infty$ structure on $M^m$, $\mathcal{F}_1=\{(U_\alpha,\varphi_\alpha)\}\in [\mathcal{F}].$ if $(V,\psi)$ is compatible with each coordinate map $C^\infty$, the map can be added to obtain the expanded coordinate atlas $\mathcal{F}_1'.$, which is also the $C^\infty$ coordinate atlas, and is still in the equivalence class $[\mathcal{F}]$. In this way, the maximum $C^\infty$ coordinate atlas $\mathcal{F}_{\max}\in [\mathcal{F}],$ can be obtained, which is also called the $C^\infty$ structure.
Right $M=\mathbb{R}^m,$
Taking $\mathcal{F}=\{(\mathbb{R}^m,\mathrm{id})\}$ is $C^\infty$
$(C^\omega)$ Coordinate atlas,
Call $[\mathcal{F}]$ the standard $C^\infty$ structure on $\mathbb{R}^m$,
$(\mathbb{R}^m,[\mathcal{F}]^{C^\infty})$ is a $m$-dimensional $C^\infty$ manifold.
The article was last updated on 2021-09-15 16:12:46
