Machine-translated from Chinese.
切丛
回忆$M^m$为$m$维$C^\infty$流形,
其上有切丛$TM=\bigcup_{p\in M}T_pM\xrightarrow{\pi}M,$ 具有自然的拓扑.
$T_qM$有自然基$\{X_i\},$ $T_qM\cong \mathbb{R}^m,$
从而有同胚$\Phi_U:\pi^{-1}(U)\rightarrow U\times \mathbb{R}^m\rightarrow \varphi(U)\times \mathbb{R}^m.$
$\{(\pi^{-1}(U_\alpha),\Phi_\alpha)\}$即为$TM$的光滑坐标图册,
使之成为$2m$维$C^\infty$流形.
向量场
$X:p\mapsto X_p\in T_pM$称为$M$上向量场. 对开集$U\subset M,$
$X:U\ni p\mapsto T_pM$称为$U$上向量场.
也可以描述$X$为$X:M\rightarrow TM,$ $\pi\circ X=\mathrm{id}_M.$
如果$X$还是流形间的$C^\infty$映照, 则称$X$是$M$上$C^\infty$向量场.
考虑局部拉回到$\widehat{X}:\varphi(U)\rightarrow \varphi(U)\times \mathbb{R}^m$,
具体表示为$x\mapsto (x,\xi\circ\varphi^{-1}(x)).$
因此只需考虑$\xi\circ\varphi^{-1}$是否光滑即可,
也即$\xi$各个分量是否在$C^\infty(U)$中.
用$\chi(M)$表示$M$山光滑向量场全体, $\,\forall\,X,Y\in \chi(M),$
$f,g\in C^\infty(M),$ 则$fX+gY\in \chi(M),$
$(fX+gY)(p):=f(p)X(p)+g(p)Y(p)\in T_pM.$
这是因为$C^\infty(U)$和$C^\infty(\mathbb{R}^m)$一样,
都是乘法封闭的向量空间, 考虑局部表示即可.
特别地$\chi(M)$是无穷维向量空间.
回忆若$f\in C^2(U),$ $U\subset \mathbb{R}^m,$ $f$的混合偏导可交换顺序.
那么对一般的向量场, 我们考虑是否有$XY=YX,$ $X,Y\in\chi(M).$
即$[X,Y]f=XYf-YXf$是否总取零.
这里$[-,-]:\chi(M)\times \chi(M)\rightarrow\chi(M)$称为$X$和$Y$的Lie括号.
$[X,Y]_p f=X_pYf-Y_pXf.$
回忆$(Yf)(p)=Y_p(f),$ 希望$Yf\in C^\infty(M).$
考察$(Yf)\circ\varphi^{-1},$ $(Yf)|_U=\sum_i\eta^i X_i(f),$
那么$(Yf)\circ\varphi^{-1}=\sum_i \eta^i\circ\varphi^{-1}\cdot \frac{\partial f\circ\varphi^{-1} }{\partial x_i}\in C^\infty(\mathbb{R}^m).$
因此$X_pYf$有意义, 有:
$$\begin{aligned}
X_pYf&=\sum_i \xi^i(p)\sum_j X_i(\eta^jX_j(f))\\
&=\sum_{i,j} \xi^i(p)X_i(\eta^j)X_j(f)+\xi^i(p)\eta^j(p)X_iX_j(f)
\end{aligned}$$
交换$X,Y$顺序即得到$Y_pXf,$ 从而:
$$[X,Y]_pf=\sum_{i,j}[\xi^i(p)X_i(\eta^j)-\eta^i(p)X_i(\xi^j)]X_j(f)+\sum_{i,j}\xi^i(p)\eta^j(p)[X_i,X_j] (f),$$
后项由于$f\in C^\infty(M)$取零,
故$[X,Y]|_U=\sum_{ij} [\xi^iX_j(\eta^j)-\eta^i X_i(\xi^j)]X_j,$
$[X,Y]\in\chi(M)$确实为光滑向量场.
可以看出, $[-,-]$满足如下性质:
反交换性: $[X,Y]=-[Y,X];$
线性性: $[aX_1+bX_2,Y]=a[X_1,Y]+b[X_2,Y];$
Jacobi恒等式: $[[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0.$
因此$\chi(M)$是一个李代数.
切映照
设$F:M^m\rightarrow N^n$是$C^\infty$流形间的$C^\infty$映照.
$\,\forall\,$过$p$点曲线$C(t),$ $C(t_0)=p,$ $C'(t_0)=X,$
则对应有过$q$点曲线$F(C(t)),$ 切向量为$Y.$
这诱导了切空间之间的映照${F_\ast }_p:T_pM\rightarrow T_qN.$
$\,\forall\,f\in C_q^\infty(N),$ $Yf=X(f\circ F).$
即${F_\ast }_pX(f)=X(f\circ F).$ 易证
${F_\ast }_p(aX_1+bX_2)=a{F_\ast }_pX_1+b{F_\ast }_pX_2,$ 因此${F_\ast }_p$为线性映照.
回忆向量空间之间的线性映照$\varphi:V\rightarrow W$诱导对偶映照$\varphi^\ast :W^\ast \rightarrow V^\ast ,$
定义为$\,\forall\,w^\ast \in W^\ast ,$ $(\varphi^\ast w^\ast )(v)=w^\ast (\varphi (v)).$
对偶映照也是线性的, 称为拉回映照. 因此对上面的${F_\ast }_p,$
有拉回映照$F^\ast :T_q^\ast N\rightarrow T_p^\ast M,$
$F^\ast (df)_q(X)=(df)_q({F_\ast }_pX)={F_\ast }_pX(f)=X(f\circ F)= d(f\circ F)_p(X),$
即$F^\ast (df)_q=d(f\circ F)_p.$
命题 1. 设$F\in C^\infty(M,N),$ $G\in C^\infty(N,W),$ 则${(G\circ F)_{\ast } }_p={G_{\ast } }_q\circ {F_\ast }_p,$ $(G\circ F)^\ast =F^\ast \circ G^\ast .$ 且若$F$是光滑同胚, 那么${F_\ast }_p$是线性同构, $\,\forall\,p\in M.$
前面的性质是容易说明的, 而后面的性质利用$F^{-1}\circ F=\mathrm{id}_M,$
$F\circ F^{-1}=\mathrm{id}_N,$ 结合前面的性质即得.
命题 2. 设$F:M\rightarrow N$是光滑映照, $q=F(p).$ 那么${F_\ast }_pX_i=\sum_\alpha [X_i(y^\alpha \circ F)]Y_\alpha,$ $F^\ast (dy^\alpha)_q=\sum_i [X_i(y^\alpha\circ F)] (dx^i)_p.$
按定义验证即可.
注意$X_i(y^\alpha\circ F)=J(\psi\circ F\circ \varphi^{-1})_{\alpha,i},$
因此切映照是Jacobi矩阵的推广.
给出如下定义:
定义$F$在$p$点的秩为${F_{\ast } }_p$的秩$=r(D\widehat{F})_{\varphi(p)},$
记为$r({F_\ast }_p);$
如果${F_\ast }_p$是单射, 即$r({F_\ast }_p)=m,$ 则称$F$在$p$点是浸入.
若$F$点点浸入, 则称$F:M\rightarrow N$是浸入;
如果${F_\ast }_p$是满射, 即$r({F_\ast }_p)=n,$ 则称$F$在$p$点是淹没.
若$F$点点淹没, 则称$F:M\rightarrow N$是淹没.
回忆反函数定理: 若开集$U,V\subset \mathbb{R}^m,$
$F:U\rightarrow V$为光滑映照, $p\in U,$ $q=F(p)\in V.$ 若$DF(p)$非退化,
那么$\,\exists\,$点$p$的小邻域$U_1,$ $q$点小邻域$V_1,$
使$F|_{U_1}:U_1\rightarrow V_1$是$C^\infty$同胚. 那么在流形上,
设$U\subset M^m,V\subset N^m,$
将$DF(p)$非退化条件改为${F_\ast }_p$是线性同构, 即可推广反函数定理到流形上.
文章最后更新于 2021-11-21 17:57:10
Tangent Bundle
Recall that $M^m$ is a $m$-dimensional $C^\infty$ manifold,
There is a tangent bundle $TM=\bigcup_{p\in M}T_pM\xrightarrow{\pi}M,$ on it with natural topology.
$T_qM$ Has natural base $\{X_i\},$ $T_qM\cong \mathbb{R}^m,$
Thus there is homeomorphism$\Phi_U:\pi^{-1}(U)\rightarrow U\times \mathbb{R}^m\rightarrow \varphi(U)\times \mathbb{R}^m.$
$\{(\pi^{-1}(U_\alpha),\Phi_\alpha)\}$ is the smooth coordinate atlas of $TM$,
Make it a $2m$-dimensional $C^\infty$ manifold.
vector field
$X:p\mapsto X_p\in T_pM$ is called $M$ on vector field. Folio set$U\subset M,$
$X:U\ni p\mapsto T_pM$ is called the vector field on $U$.
$X$ can also be described as $X:M\rightarrow TM,$ $\pi\circ X=\mathrm{id}_M.$
If $X$ is also the mapping of $C^\infty$ between manifolds, then $X$ is said to be the $C^\infty$ vector field on $M$.
Consider partially pulling back to $\widehat{X}:\varphi(U)\rightarrow \varphi(U)\times \mathbb{R}^m$,
Specifically expressed as $x\mapsto (x,\xi\circ\varphi^{-1}(x)).$
Therefore, you only need to consider whether $\xi\circ\varphi^{-1}$ is smooth,
That is, whether each component of $\xi$ is in $C^\infty(U)$.
Use $\chi(M)$ to represent the entire $M$ mountain smooth vector field, $\,\forall\,X,Y\in \chi(M),$
$f,g\in C^\infty(M),$ then $fX+gY\in \chi(M),$
$(fX+gY)(p):=f(p)X(p)+g(p)Y(p)\in T_pM.$
This is because $C^\infty(U)$ is the same as $C^\infty(\mathbb{R}^m)$,
They are all multiplicatively closed vector spaces, so just consider local representation.
In particular, $\chi(M)$ is an infinite-dimensional vector space.
Recall that the order of mixed partial derivatives of $f\in C^2(U),$ $U\subset \mathbb{R}^m,$ $f$ can be exchanged.
So for a general vector field, we consider whether there is $XY=YX,$ $X,Y\in\chi(M).$
That is, whether $[X,Y]f=XYf-YXf$ always takes zero.
Here $[-,-]:\chi(M)\times \chi(M)\rightarrow\chi(M)$ is called $X$ and $Y$ Lie brackets.
$[X,Y]_p f=X_pYf-Y_pXf.$
Memories$(Yf)(p)=Y_p(f),$ Hope$Yf\in C^\infty(M).$
Inspection $(Yf)\circ\varphi^{-1},$ $(Yf)|_U=\sum_i\eta^i X_i(f),$
Then $(Yf)\circ\varphi^{-1}=\sum_i \eta^i\circ\varphi^{-1}\cdot \frac{\partial f\circ\varphi^{-1} }{\partial x_i}\in C^\infty(\mathbb{R}^m).$
Therefore $X_pYf$ makes sense, with:
$$\begin{aligned}
X_pYf&=\sum_i \xi^i(p)\sum_j X_i(\eta^jX_j(f))\\
&=\sum_{i,j} \xi^i(p)X_i(\eta^j)X_j(f)+\xi^i(p)\eta^j(p)X_iX_j(f)
\end{aligned}$$
Swap the order of $X,Y$ to get $Y_pXf,$ thus:
$$[X,Y]_pf=\sum_{i,j}[\xi^i(p)X_i(\eta^j)-\eta^i(p)X_i(\xi^j)]X_j(f)+\sum_{i,j}\xi^i(p)\eta^j(p)[X_i,X_j] (f),$$
The latter term is zero due to $f\in C^\infty(M)$,
Therefore$[X,Y]|_U=\sum_{ij} [\xi^iX_j(\eta^j)-\eta^i X_i(\xi^j)]X_j,$
$[X,Y]\in\chi(M)$ is indeed a smooth vector field.
It can be seen that $[-,-]$ satisfies the following properties:
Anti-commutativity: $[X,Y]=-[Y,X];$
Linearity: $[aX_1+bX_2,Y]=a[X_1,Y]+b[X_2,Y];$
Jacobi identity: $[[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0.$
Therefore $\chi(M)$ is a Lie algebra.
cut reflection
Let $F:M^m\rightarrow N^n$ be the $C^\infty$ mapping between $C^\infty$ manifolds.
$\,\forall\,$ Curve through $p$ point $C(t),$ $C(t_0)=p,$ $C'(t_0)=X,$
Then the tangent vector of the curve $F(C(t)),$ corresponding to the point $q$ is $Y.$
This induces mapping between tangent spaces ${F_\ast }_p:T_pM\rightarrow T_qN.$
$\,\forall\,f\in C_q^\infty(N),$ $Yf=X(f\circ F).$
That is ${F_\ast }_pX(f)=X(f\circ F).$ Easy to prove
${F_\ast }_p(aX_1+bX_2)=a{F_\ast }_pX_1+b{F_\ast }_pX_2,$ Therefore ${F_\ast }_p$ is a linear mapping.
Recall a linear mapping between vector spaces $\varphi:V\rightarrow W$ and induce a dual mapping $\varphi^\ast :W^\ast \rightarrow V^\ast ,$
Defined as $\,\forall\,w^\ast \in W^\ast ,$ $(\varphi^\ast w^\ast )(v)=w^\ast (\varphi (v)).$
The dual mapping is also linear and is called Pullback map. Therefore for the above ${F_\ast }_p,$
With pullback map $F^\ast :T_q^\ast N\rightarrow T_p^\ast M,$
$F^\ast (df)_q(X)=(df)_q({F_\ast }_pX)={F_\ast }_pX(f)=X(f\circ F)= d(f\circ F)_p(X),$
That is $F^\ast (df)_q=d(f\circ F)_p.$
Proposition 1. Assume $F\in C^\infty(M,N),$ $G\in C^\infty(N,W),$ then ${(G\circ F)_{\ast } }_p={G_{\ast } }_q\circ {F_\ast }_p,$ $(G\circ F)^\ast =F^\ast \circ G^\ast .$ and if $F$ is a smooth homeomorphism, then ${F_\ast }_p$ is a linear isomorphism, $\,\forall\,p\in M.$
The former properties are easy to explain, while the latter properties use $F^{-1}\circ F=\mathrm{id}_M,$
$F\circ F^{-1}=\mathrm{id}_N,$ It can be obtained by combining the previous properties.
Proposition 2. Assume $F:M\rightarrow N$ is a smooth mapping, $q=F(p).$ then ${F_\ast }_pX_i=\sum_\alpha [X_i(y^\alpha \circ F)]Y_\alpha,$ $F^\ast (dy^\alpha)_q=\sum_i [X_i(y^\alpha\circ F)] (dx^i)_p.$
Just verify according to the definition.
Note$X_i(y^\alpha\circ F)=J(\psi\circ F\circ \varphi^{-1})_{\alpha,i},$
Therefore the cut mapping is a generalization of the Jacobi matrix.
Give the following definition:
Define $F$ at point $p$ rank is the rank $=r(D\widehat{F})_{\varphi(p)},$ of ${F_{\ast } }_p$
Recorded as $r({F_\ast }_p);$
If ${F_\ast }_p$ is an injective, that is, $r({F_\ast }_p)=m,$, then $F$ at point $p$ is said to be Dip.
If $F$ is immersed little by little, then $F:M\rightarrow N$ is said to be immersed;
If ${F_\ast }_p$ is surjective, that is, $r({F_\ast }_p)=n,$, then $F$ is said to be at point $p$ submerged.
If $F$ is submerged little by little, then $F:M\rightarrow N$ is said to be submerged.
Recall the inverse function theorem: Ifan set $U,V\subset \mathbb{R}^m,$
$F:U\rightarrow V$ is smooth mapping, $p\in U,$ $q=F(p)\in V.$ if $DF(p)$ is not degenerate,
Then $\,\exists\,$ points to the small neighborhood $U_1,$ of $p$ $q$ and points to the small neighborhood $V_1,$
Let $F|_{U_1}:U_1\rightarrow V_1$ be homeomorphic to $C^\infty$. Then on the manifold,
Let $U\subset M^m,V\subset N^m,$
By changing $DF(p)$ non-degenerate condition to ${F_\ast }_p$ which is linear isomorphism, the inverse function theorem can be generalized to manifolds.
The article was last updated on 2021-11-21 17:57:10
