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子流形
设$F:M^m\hookrightarrow N^n$为子流形. 若$\,\forall\,p\in M,$
存在含$q=F(p)$的$N$中坐标系$(V,\psi,y^\alpha),$
使得$F(M)\cap V$为$V$中切片, 则称$(M,F)$为$N$中的正则子流形.
另外, $F(M)$是$N$的子集, 可以从$N$中诱导子空间拓扑,
也可以通过$F$将$M$上的拓扑搬到$F(M)$上. 若两种诱导拓扑一致,
则称$(M,F)$为嵌入子流形.
定理 1.1. 设$F:M\rightarrow N$是子流形, 则$(M,F)$是正则子流形当且仅当它是嵌入子流形.
证明可参考白,沈的《黎曼几何初步》.
定理 1.2. 设$F:M\rightarrow N$是子流形. 如果$M$紧致, 则$(M,F)$一定是嵌入子流形.
证: 赋予$F(M)$作为$N$中子集所诱导的拓扑$\mathcal{T},$
只需说明$F:M\rightarrow F(M)$为同胚. 首先说明$F$是连续的.
对任意开集$U\subset F(M),$ 存在开集$\widetilde{U}\subset N,$
使得$U=\widetilde{U}\cap F(M).$
那么$F^{-1}(U)=F^{-1}(\widetilde{U})$是$M$中开集. 接下来说明$F$为闭映射.
而紧空间到$T_2$空间的连续映射都是闭映射, 由此得证.
定理 1.3. 设$F:M^m\rightarrow N^n$是光滑映照. 取$q\in N,$ 定义$P:=F^{-1}(q).$ 若$\,\forall\,p\in P,$ $r({F_{\ast } }_p)=n,$ 则$P$具有唯一的$C^\infty$-流形结构使$(P,i)$是$M$的正则子流形, $\dim P=m-n.$
证: 由秩定理, 可以取到合适的盒形坐标系使$\widehat{F}$为投影.
因此$\widehat{F}^{-1}(0)=\{(0,\cdots,0,x^{n+1},\cdots,x^m)||x^\gamma|<\delta\},$
即$\widehat{F}^{-1}(0)=F^{-1}(q)\cap U$为$U$中$n$维切片.
赋予$P$子空间拓扑, 取坐标图册$\{(P\cap U,\pi\circ\varphi)\},$
易见$P$成为$m-n$维拓扑流形. 同时容易看出图册满足$C^\infty$相容性,
因此$P$具有$C^\infty$-流形结构. 显然单射$i\in C^\infty(P,M)$且秩为$m-n,$
因此它是子流形. 由先前说明, 局部来看它又是切片, 因此$(P,i)$是正则子流形.
推论 1.4. 如果满射$F:M\rightarrow N$是淹没, 则$\,\forall\,q\in N,$ $F^{-1}(q)$为$M$中$m-n$维正则子流形.
一个问题是给定$C^\infty$流形$M^m,$
是否存在$F:M\rightarrow \mathbb{R}^N,$
使$(M,F)$是$\mathbb{R}^N$中子流形. H.Whitney 指出,
存在$F:M^m\hookrightarrow \mathbb{R}^{2m+1}$为正则子流形.
向量场的积分曲线
设$X\in \chi(M),$ $\gamma:(a,b)\rightarrow M$为$C^\infty$-曲线,
若$\gamma'(t)=X_{\gamma(t)},$ $\,\forall\,t\in (a,b).$
则称$\gamma$是$X$的积分曲线. 如果$0\in(a,b),$
$p=\gamma(0)$称为$\gamma$的初始点.
给定$X\in\chi(M),$ $p\in M.$ 任取坐标系$(U,\varphi;x^i),$
$\{X_i\}_{i=1}^m$为自然基. 那么$X|_U=\sum \xi^iX_i,$
$\xi^i\in C^\infty(U).$ 而$\gamma'(t)=\sum_i \frac{dx^i(t)}{dt}X_i,$
因此求积分曲线$\gamma$即在给定初值后解常微分方程组:
$$\frac{dx^i}{dt}=\xi^i(\gamma(t))=\widehat{\xi}^i(x^1,\cdots,x^m),\quad i=1,\cdots,m.$$
其中$\widehat{\xi}^i=\xi^i\circ\varphi^{-1}\in C^\infty(\varphi(U)).$
由常微分方程基本定理, 我们有如下命题:
命题 1.5. 设$X\in\chi(M),$ 则$\,\forall\,p\in M,$ $\,\exists\,\varepsilon>0,$ $\gamma:(-\varepsilon,\varepsilon)\rightarrow M,$ 使得$\gamma(0)=p$且$\gamma$为$X$的积分曲线.
设$X\in\chi(M),$ 若$X_p=0,$ 则称$p$为$X$的奇点. 一个经典的命题是,
球面上光滑向量场必有奇点. 一个典型的证明是应用指标定理,
而球面的欧拉示性数非零.
回忆切丛$TM$为$2m$维$C^\infty$流形, 对$U\subset M,$
有局部平凡化$\widetilde{\Phi}:TU=\pi^{-1}(U)\cong U\times \mathbb{R}^m$为微分同胚,
且$\widetilde{\Phi}|_{T_{p}M}:T_pM\rightarrow \{p\}\times \mathbb{R}^m$为线同构(保纤维).
那么上面的命题说明, $TS^2\not\cong S^2\times \mathbb{R}^2,$
即球面切丛非平凡丛.
文章最后更新于 2021-10-27 21:07:04
submanifold
Let $F:M^m\hookrightarrow N^n$ be a submanifold. If $\,\forall\,p\in M,$
There is a coordinate system $(V,\psi,y^\alpha),$ in $N$ containing $q=F(p)$
If $F(M)\cap V$ is a slice in $V$, then $(M,F)$ is called a slice in $N$ regular submanifold.
In addition, $F(M)$ is a subset of $N$, and the subspace topology can be induced from $N$,
You can also move the topology on $M$ to $F(M)$ through $F$. If the two induced topologies are consistent,
Then $(M,F)$ is called embedded submanifold.
Theorem 1.1. Assume $F:M\rightarrow N$ is a submanifold, then $(M,F)$ is a regular submanifold if and only if it is an embedded submanifold.
For proof, please refer to "Preliminary Riemannian Geometry" by Bai and Shen.
Theorem 1.2. Let $F:M\rightarrow N$ be a submanifold. If $M$ is compact, then $(M,F)$ must be an embedded submanifold.
Certificate: Assign $F(M)$ to the topology $\mathcal{T},$ induced by a subset of $N$
We only need to explain that $F:M\rightarrow F(M)$ is homeomorphism. First, explain that $F$ is continuous.
For any open set $U\subset F(M),$, there exists an open set $\widetilde{U}\subset N,$
Make$U=\widetilde{U}\cap F(M).$
Then $F^{-1}(U)=F^{-1}(\widetilde{U})$ is an open set in $M$. Next, $F$ is a closed map.
The continuous mapping from compact space to $T_2$ space is a closed mapping, which is proved by this.
Theorem 1.3. Let $F:M^m\rightarrow N^n$ be a smooth mapping. Take $q\in N,$ and define $P:=F^{-1}(q).$. If $\,\forall\,p\in P,$ $r({F_{\ast } }_p)=n,$, then $P$ has a unique $C^\infty$-manifold structure such that $(P,i)$ is a regular submanifold of $M$, $\dim P=m-n.$
Certificate: According to the rank theorem, a suitable box coordinate system can be obtained so that $\widehat{F}$ is the projection.
Therefore $\widehat{F}^{-1}(0)=\{(0,\cdots,0,x^{n+1},\cdots,x^m)||x^\gamma|<\delta\},$
That is, $\widehat{F}^{-1}(0)=F^{-1}(q)\cap U$ is the $n$-dimensional slice of $U$.
Give $P$ subspace topology, get coordinate atlas $\{(P\cap U,\pi\circ\varphi)\},$
It is easy to see that $P$ becomes a $m-n$-dimensional topological manifold. At the same time, it is easy to see that the atlas satisfies the $C^\infty$ compatibility,
Therefore $P$ has a $C^\infty$-manifold structure. Obviously it is injective $i\in C^\infty(P,M)$ with rank $m-n,$
Therefore it is a submanifold. From the previous explanation, it is also a slice locally, so $(P,i)$ is a regular submanifold.
Corollary 1.4. If the surjection $F:M\rightarrow N$ is submerged, then $\,\forall\,q\in N,$ $F^{-1}(q)$ is the $m-n$-dimensional regular submanifold in $M$.
A problem is given $C^\infty$ manifold $M^m,$
Does it exist$F:M\rightarrow \mathbb{R}^N,$
Let $(M,F)$ be the $\mathbb{R}^N$ neutron manifold. H.Whitney pointed out,
There exists $F:M^m\hookrightarrow \mathbb{R}^{2m+1}$ as a regular submanifold.
Integral curve of vector field
Let $X\in \chi(M),$ $\gamma:(a,b)\rightarrow M$ be $C^\infty$-curve,
If $\gamma'(t)=X_{\gamma(t)},$ $\,\forall\,t\in (a,b).$
Then $\gamma$ is said to be of $X$ integral curve.if$0\in(a,b),$
$p=\gamma(0)$ is called $\gamma$ initial point.
Given $X\in\chi(M),$ $p\in M.$ and any coordinate system $(U,\varphi;x^i),$
$\{X_i\}_{i=1}^m$ is a natural basis. Then $X|_U=\sum \xi^iX_i,$
$\xi^i\in C^\infty(U).$ and $\gamma'(t)=\sum_i \frac{dx^i(t)}{dt}X_i,$
Therefore, finding the integral curve $\gamma$ means solving the system of ordinary differential equations after given the initial value:
$$\frac{dx^i}{dt}=\xi^i(\gamma(t))=\widehat{\xi}^i(x^1,\cdots,x^m),\quad i=1,\cdots,m.$$
Among them $\widehat{\xi}^i=\xi^i\circ\varphi^{-1}\in C^\infty(\varphi(U)).$
According to the fundamental theorem of ordinary differential equations, we have the following proposition:
Proposition 1.5. Assume $X\in\chi(M),$ then $\,\forall\,p\in M,$ $\,\exists\,\varepsilon>0,$ $\gamma:(-\varepsilon,\varepsilon)\rightarrow M,$ such that $\gamma(0)=p$ and $\gamma$ are the integral curves of $X$.
Assume $X\in\chi(M),$ if $X_p=0,$, then $p$ is called $X$ Singularity. A classic proposition is,
A smooth vector field on a sphere must have singular points. A typical proof is to apply the index theorem,
The Euler characteristic number of the sphere is non-zero.
Recall that the tangent bundle $TM$ is the $2m$-dimensional $C^\infty$ manifold, for $U\subset M,$
There is local trivialization $\widetilde{\Phi}:TU=\pi^{-1}(U)\cong U\times \mathbb{R}^m$ as diffeomorphism,
And $\widetilde{\Phi}|_{T_{p}M}:T_pM\rightarrow \{p\}\times \mathbb{R}^m$ is line isomorphism (fiber preserving).
Then the above proposition explains, $TS^2\not\cong S^2\times \mathbb{R}^2,$
That is, the spherical tangent bundle is a non-trivial bundle.
The article was last updated on 2021-10-27 21:07:04
