Machine-translated from Chinese.
切空间
回忆我们可以定义$C^\infty$曲线$C:(a,b)\rightarrow M,$ $C(t_0)=p.$
$C_p^\infty$表示$M$上在$p$点$C^\infty$的函数全体,
即其中元素均是点$p$某邻域上的光滑函数. $C_p^\infty$上函数可做运算,
但是仅能在公共邻域上做运算.
曲线$C(t)$在点$p$处”切向量”指方向导数算子$X:C_p^\infty\rightarrow \mathbb{R},$
$X(f)=\frac{df\circ c(t)}{dt}|_{t=t_0},$ 满足:
(1) $X(af+bg)=aX(f)+bX(g);$
(2) $X(fg)=X(f)g(p)+f(p)X(g).$
令$T_pM:=\{X\text{是过}p\text{的曲线的切向量}\},$
称为$M$在$p$处的切空间.
$\widetilde{T}_pM:=\{\widetilde X:C_p^\infty\rightarrow \mathbb{R}\text{满足(1),(2)}\}.$
那么有$T_pM\subset \widetilde{T}_pM.$ $\widetilde{T}_pM$可引入加法,
数乘使之成为向量空间.
回忆对坐标卡$(U,\varphi),$ $\varphi$是流形间的光滑同胚.
对$\varphi^{-1},$ 仅让$x^i$变化,
即得光滑曲线$\varphi^{-1}(\cdots,x^i,\cdots),$ 称为$x^i$-曲线.
过每点$q\in U$有$m$条坐标曲线.
用$\left(\frac{\partial {} }{\partial {}x^i}\right)_p$表示$x_i$-曲线在$p$处切向量$X_i,$
$X_i(f)=\frac{d}{dx^i}[f\circ \varphi^{-1}(\cdots, x^i,\cdots)]_{\varphi(p)}=[\frac{\partial {} }{\partial {}x^i}(f\circ\varphi^{-1})]_{\varphi(p)}.$
$f\circ\varphi^{-1}:\varphi(U)\rightarrow \mathbb{R}.$
从而$\left(\frac{\partial {} }{\partial {}x^i}\right)_p\in T_pM,$
$1\le i\le m.$
命题 1.1. $T_pM$是$m$维向量空间. $\{X_i\}$构成$T_pM$的一组基.
证: 任取$X\in T_pM,$ $C:(a,b)\rightarrow M,$ $C(t_0)=p.$
$C'(t_0)=X.$ $\varphi\circ C(t)=(x^i(t))$称为参数方程. 由定义,
$X(f)=\frac{d}{dt}[f\circ C(t)]|_{t=t_0}=\frac{d}{dt}[(f\circ\varphi^{-1})(\varphi\circ C(t))]|_{t_0}=\sum_{i=1}^m \frac{dx^i(t)}{dt}|_{t_0}\cdot [\frac{\partial {} }{\partial {}x^i}(f\circ \varphi^{-1})]_{\varphi(p)}.$
记$a^i=\frac{d x^i}{dt}|_{t_0},$ 那么$X(f)=\sum_{i=1}^m a^i X_i(f),$
从而$X=\sum_{i=1}^m a^i X_i.$
任给$\sum_{i=1}^m b^i X_i,$
要找$C(t)$过$p$点使$\frac{dx^i(t)}{dt}|_{t_0}=b^i.$
设$\varphi(p)=(d^i),$ 取$x^i(t)=d^i+b^i(t-t_0),$
$C(t)=\varphi^{-1}(x^i(t))$即可. 那么$C'(t_0)=\sum_i b^iX_i.$
进而即可说明$T_pM$是向量空间.
只需再说明$\{X_i\}$线性无关. 设$\sum_{i=1}^m a^i X_i=0,$
取$m$个只沿$x^i$方向增长的$f=x^i:=\pi^i\circ \varphi$试探,
即可说明$a^i\equiv 0.$ 从而$T_pM$为$m$维向量空间.
注 1.2. 事实上, 有$T_pM=\widetilde{T}_pM.$
之前提到$C_p^\infty$中零元素不唯一, 因此并非向量空间.
为此可以定义等价关系$f\sim g$ $\Leftrightarrow$ $\,\exists\,p$点邻域$W,$
$f|_W=g|_W.$ 商空间$\mathcal{F}_p^\infty=C_p^\infty/\sim$为向量空间,
称为函数芽空间.
其上有诱导的$X:\mathcal{F}_p^\infty\rightarrow \mathbb{R},$
即$X([f]):=X(f),$ $[f](p):=f(p).$ 它同样满足类似(1),(2)的性质.
余切空间
设$V$为$m$维向量空间,
$V^\ast =\{\text{linear }\omega:V\rightarrow \mathbb{R}\}$亦为$m$维向量空间,
以对偶基为基. 对$m$维光滑流形$M^m,$ $T_pM$为切空间,
$T_p^\ast M$表示其对偶空间, 称为$M$在$p$处余切空间.
$X\in T_pM,$ $f\in C_p^\infty$有方向导数$X(f).$ 考虑固定$f$令$X$变化,
记$\omega_f(X)=X(f),$
那么$\omega_f:T_pM\rightarrow \mathbb{R}$为线性函数, 在$T_p^\ast M$中.
记$\omega_f$为$(df)_p,$ 称为$f$在$p$点的微分.
任取含$p$坐标系$(U,\varphi),$ 那么$\{(dx^i)_p\}\in T_p^\ast M,$
为$\{X_i\}$的对偶基, 因此构成$T_p^\ast M$的基, 称为自然基.
$\,\forall\,f\in C_p^\infty,$
$(df)_p=\sum_{i=1}^m a_i(dx^i)_p\in T_p^\ast M.$ 两边作用$X_j,$
得$X_j(f)=a_j.$ 因此有$(df)_p=\sum_{i=1}^m X_i(f)(dx^i)_p.$
由此也可以看出称其为微分的原因.
注 1.3. 若$f\sim g\in C_p^\infty,$ 那么$X(f)=X(g),$ 从而$(df)_p=(dg)_p.$ 那么记$\tau:\mathcal{F}_p^\infty\rightarrow T_p^\ast M,$ $\tau([f])=(df)_g$ 是线性满的. $\widehat{\tau}:\mathcal{F}_p^\infty/\ker(\tau)\cong T_p^\ast M$为线性同构, $\ker(\tau)=\{[f]\in \mathcal{F}_p^\infty|\tau([f])=(df)_p=0\}.$ 因此对余切空间的另一种定义方式就是函数芽空间做商空间, 然后再对偶来定义切空间.
切丛
取$TM=\bigcup_{p\in M}T_pM,$ 称为$M$的切丛. 引入$\pi:TM\rightarrow M,$
$\pi(T_pM)\rightarrow p,$ 称为自然投影.
任取坐标卡$(U,\varphi; x^i),$ 考察$\pi^{-1}(U).$ $\,\forall\,q\in U,$
$T_qM$有自然基, 于是$T_qM\cong \mathbb{R}^m.$
从而有$$\widehat{\Phi}:\pi^{-1}(U)\xrightarrow{\Phi}U\times \mathbb{R}^m\xrightarrow{ \varphi\times \mathrm{id}_{\mathbb{R}^m} }\varphi(U)\times \mathbb{R}^m,$$
其中$\Phi(X)=(\pi(X),\xi),$ $X=\sum_{i=1}^m \xi^i X_i.$
赋予$TM$自然拓扑结构, 使得$\pi^{-1}(U)$是$TM$的开子集,
且$\widehat\Phi:\pi^{-1}(U)\rightarrow \varphi(U)\times \mathbb{R}^m$为同胚.
令所有$\pi^{-1}(U_i)$的开子集构成生成$TM$的拓扑基即可.
那么$TM$成为$2m$维拓扑流形.
此时$\widehat{\mathcal{F} }=[\{(\pi^{-1}(U_\alpha),\widehat{\Phi}_\alpha)\}]$为$TM$坐标图册,
转移函数$\widehat{\Phi}_\beta\circ \widehat{\Phi}_\alpha^{-1}(x,\xi)=(\varphi_{\beta}\circ\varphi_{\alpha}^{-1}(x),J(\varphi_\beta\circ\varphi_\alpha^{-1})\xi)$为光滑函数,
因此图册是光滑的, 故切丛$TM$为$2m$维$C^\infty$流形.
文章最后更新于 2021-10-09 12:26:27
Tangent Space
Recall that we can define $C^\infty$ curve $C:(a,b)\rightarrow M,$ $C(t_0)=p.$
$C_p^\infty$ represents the entire function on $M$ at $p$ point $C^\infty$,
That is, the elements are all smooth functions on a certain neighborhood of point $p$. The function on $C_p^\infty$ can perform operations,
But operations can only be performed on common neighborhoods.
The "tangent vector" of curve $C(t)$ at point $p$ points to the directional derivative operator $X:C_p^\infty\rightarrow \mathbb{R},$
$X(f)=\frac{df\circ c(t)}{dt}|_{t=t_0},$ Satisfy:
(1) $X(af+bg)=aX(f)+bX(g);$
(2) $X(fg)=X(f)g(p)+f(p)X(g).$
Let$T_pM:=\{X\text{是过}p\text{的曲线的切向量}\},$
It is called the tangent space of $M$ at $p$.
$\widetilde{T}_pM:=\{\widetilde X:C_p^\infty\rightarrow \mathbb{R}\text{满足(1),(2)}\}.$
Then there are $T_pM\subset \widetilde{T}_pM.$ $\widetilde{T}_pM$ that can introduce addition,
Multiplying numbers makes it a vector space.
Recall that the coordinate pairs $(U,\varphi),$ $\varphi$ are smooth homeomorphisms between manifolds.
For $\varphi^{-1},$, only $x^i$ changes,
That is, the smooth curve $\varphi^{-1}(\cdots,x^i,\cdots),$ is obtained, which is called $x^i$-curve.
There are $m$ coordinate curves passing through each point $q\in U$.
Use $\left(\frac{\partial {} }{\partial {}x^i}\right)_p$ to represent $x_i$ - the curve tangent vector $X_i,$ at $p$
$X_i(f)=\frac{d}{dx^i}[f\circ \varphi^{-1}(\cdots, x^i,\cdots)]_{\varphi(p)}=[\frac{\partial {} }{\partial {}x^i}(f\circ\varphi^{-1})]_{\varphi(p)}.$
$f\circ\varphi^{-1}:\varphi(U)\rightarrow \mathbb{R}.$
Thus $\left(\frac{\partial {} }{\partial {}x^i}\right)_p\in T_pM,$
$1\le i\le m.$
Proposition 1.1. $T_pM$ is a $m$-dimensional vector space. $\{X_i\}$ constitutes a set of basis of $T_pM$.
Certificate: Take any $X\in T_pM,$ $C:(a,b)\rightarrow M,$ $C(t_0)=p.$
$C'(t_0)=X.$ $\varphi\circ C(t)=(x^i(t))$ is called parametric equations. By definition,
$X(f)=\frac{d}{dt}[f\circ C(t)]|_{t=t_0}=\frac{d}{dt}[(f\circ\varphi^{-1})(\varphi\circ C(t))]|_{t_0}=\sum_{i=1}^m \frac{dx^i(t)}{dt}|_{t_0}\cdot [\frac{\partial {} }{\partial {}x^i}(f\circ \varphi^{-1})]_{\varphi(p)}.$
Remember $a^i=\frac{d x^i}{dt}|_{t_0},$ then $X(f)=\sum_{i=1}^m a^i X_i(f),$
Thus $X=\sum_{i=1}^m a^i X_i.$
Let it go $\sum_{i=1}^m b^i X_i,$
To find $C(t)$ past $p$, use $\frac{dx^i(t)}{dt}|_{t_0}=b^i.$
Let $\varphi(p)=(d^i),$ take $x^i(t)=d^i+b^i(t-t_0),$
$C(t)=\varphi^{-1}(x^i(t))$ is enough. Then $C'(t_0)=\sum_i b^iX_i.$
It can then be shown that $T_pM$ is a vector space.
We only need to explain that $\{X_i\}$ is linearly independent. Let $\sum_{i=1}^m a^i X_i=0,$
Take $m$ $f=x^i:=\pi^i\circ \varphi$ trials that only grow in the $x^i$ direction,
It can be explained that $a^i\equiv 0.$ and $T_pM$ are $m$-dimensional vector space.
Note 1.2. In fact, there is $T_pM=\widetilde{T}_pM.$
As mentioned before, the zero element in $C_p^\infty$ is not unique, so it is not a vector space.
For this purpose, the equivalence relationship $f\sim g$ $\Leftrightarrow$ $\,\exists\,p$ point neighborhood $W,$ can be defined
$f|_W=g|_W.$ Quotient space $\mathcal{F}_p^\infty=C_p^\infty/\sim$ is a vector space,
called function bud space.
There is an induced $X:\mathcal{F}_p^\infty\rightarrow \mathbb{R},$ on it
That is $X([f]):=X(f),$ $[f](p):=f(p).$ which also satisfies properties similar to (1) and (2).
cotangent space
Let $V$ be the $m$-dimensional vector space,
$V^\ast =\{\text{linear }\omega:V\rightarrow \mathbb{R}\}$ is also a $m$-dimensional vector space,
Taking the dual basis as the basis. For the $m$-dimensional smooth manifold $M^m,$ $T_pM$ as the tangent space,
$T_p^\ast M$ represents its dual space, which is called $M$ at $p$ cotangent space.
$X\in T_pM,$ $f\in C_p^\infty$ has directional derivative $X(f).$. Consider fixing $f$ and changing $X$,
Note$\omega_f(X)=X(f),$
Then $\omega_f:T_pM\rightarrow \mathbb{R}$ is a linear function, in $T_p^\ast M$.
Denote $\omega_f$ as $(df)_p,$ and call $f$ at point $p$. Differential.
Take any coordinate system $(U,\varphi),$ containing $p$, then $\{(dx^i)_p\}\in T_p^\ast M,$
is the dual basis of $\{X_i\}$, so the basis that constitutes $T_p^\ast M$ is called natural base.
$\,\forall\,f\in C_p^\infty,$
$(df)_p=\sum_{i=1}^m a_i(dx^i)_p\in T_p^\ast M.$ Both sides function $X_j,$
We get $X_j(f)=a_j.$, so we have $(df)_p=\sum_{i=1}^m X_i(f)(dx^i)_p.$
From this we can also see the reason why it is called differential.
Note 1.3. If $f\sim g\in C_p^\infty,$ then $X(f)=X(g),$ and thus $(df)_p=(dg)_p.$ then $\tau:\mathcal{F}_p^\infty\rightarrow T_p^\ast M,$ $\tau([f])=(df)_g$ is linearly complete. $\widehat{\tau}:\mathcal{F}_p^\infty/\ker(\tau)\cong T_p^\ast M$ is linear isomorphism, $\ker(\tau)=\{[f]\in \mathcal{F}_p^\infty|\tau([f])=(df)_p=0\}.$ therefore another way to define the cotangent space is to make the quotient space of the function bud space, and then use duality to define the tangent space.
Tangent Bundle
Take $TM=\bigcup_{p\in M}T_pM,$ and call it the tangent bundle of $M$. Introduce $\pi:TM\rightarrow M,$
$\pi(T_pM)\rightarrow p,$ is called natural projection.
Take any coordinate card $(U,\varphi; x^i),$ and inspect $\pi^{-1}(U).$ $\,\forall\,q\in U,$
$T_qM$ has natural basis, so $T_qM\cong \mathbb{R}^m.$
Thus there is $$\widehat{\Phi}:\pi^{-1}(U)\xrightarrow{\Phi}U\times \mathbb{R}^m\xrightarrow{ \varphi\times \mathrm{id}_{\mathbb{R}^m} }\varphi(U)\times \mathbb{R}^m,$$
Among them $\Phi(X)=(\pi(X),\xi),$ $X=\sum_{i=1}^m \xi^i X_i.$
Give $TM$ a natural topological structure such that $\pi^{-1}(U)$ is an open subset of $TM$,
And $\widehat\Phi:\pi^{-1}(U)\rightarrow \varphi(U)\times \mathbb{R}^m$ is a homeomorphism.
Let all open subsets of $\pi^{-1}(U_i)$ form the topological basis for generating $TM$.
Then $TM$ becomes a $2m$-dimensional topological manifold.
At this time $\widehat{\mathcal{F} }=[\{(\pi^{-1}(U_\alpha),\widehat{\Phi}_\alpha)\}]$ is $TM$ coordinate atlas,
The transfer function $\widehat{\Phi}_\beta\circ \widehat{\Phi}_\alpha^{-1}(x,\xi)=(\varphi_{\beta}\circ\varphi_{\alpha}^{-1}(x),J(\varphi_\beta\circ\varphi_\alpha^{-1})\xi)$ is a smooth function,
Therefore, the atlas is smooth, so the tangent bundle $TM$ is a $2m$-dimensional $C^\infty$ manifold.
The article was last updated on 2021-10-09 12:26:27
