Machine-translated from Chinese.
基本概念
对于纤维丛
$$
F\rightarrow E\rightarrow B,
$$
若$F$是$n$维(实)向量空间,
且局部平凡化还是保持向量空间结构的, 则称其为秩$n$(实)向量丛. 切丛, 法丛,
平凡向量丛丛都是典型的向量丛.
$\mathbb{R}\mathrm{P}^n$上有一个典型的向量丛, 是线丛. 定义为:
$$
E(\gamma_n^1):=\{(l,v)\in \mathbb{R}\mathrm{P}^n\times \mathbb{R}^{n+1}|v\in l\}.
$$
记$\xi$为向量丛$F\rightarrow E\xrightarrow{\pi}B.$
定义$\xi$上的截面为映射$s:B\rightarrow E,$
使得$B\xrightarrow{s}E\xrightarrow{\pi}B$为恒等映射.
称同一底空间$B$上的向量丛$\xi\cong \eta$为同构,
若$\,\exists\,$同胚$f:E(\xi)\rightarrow E(\eta),$
使得$\,\forall\,b\in B,$
$f|_{F_b(\xi)}:F_b(\xi)\cong F_b(\eta)$为向量空间间的同构.
命题 1. $\,\forall\,n\ge 1,$ $\gamma_n^1$没有非平凡处处不消失的截面. 因此$\gamma_n^1$不同构于平凡丛.
设$s:\mathbb{R}\mathrm{P}^n\rightarrow E(\gamma_n^1)$为一个截面, 考虑
$$
\mathbb{R}^{n+1}\setminus\{0\}\rightarrow \mathbb{R}\mathrm{P}^n\xrightarrow{s} E(\gamma_n^1),
$$
$$
v\mapsto ([v],t(v)v).
$$
然而$v,-v$生成同一处截面, 因此$t(-v)=-t(v).$
设$t(v)>0,$ 则$t(-v)<0,$
那么由$\mathbb{R}^{n+1}\setminus \{0\}$的连通性, 必有点$w$使得$t(w)=0,$
故截面在$[w]$处消失. (介值定理)
命题 2. 秩$n$向量丛$\xi$平凡当且仅当$\xi$上存在$n$个截面, 处处线性无关.
平凡丛找截面平凡. 假设存在这样的$n$个截面,
定义映射$B\times \mathbb{R}^n\xrightarrow{f} E(\xi),$
$$
\left(b,\sum t_ie_i\right)\mapsto \sum t_is_i(b)
$$
这是一个向量丛间的同构.
向量丛的构造
诱导丛(拉回丛)
给定向量丛$\xi,$ 连续映照$B_1\rightarrow B,$ 那么
$$
f^\ast E:=\{(b,e)\in B_1\times E|f(b)=\pi(e)\}.
$$
也就是对于$b,$
把$E$上的$f(b)$处的纤维拉回.
定义$\eta,\xi$间的丛映射为连续映照$E(\eta)\xrightarrow{g}E(\xi),$
使得它保持纤维, 且在纤维上是线性同构.
引理 3. 给定如上所述的丛映射, 我们有$\eta\cong (g|_{B(\eta)})^\ast \xi.$
只需验证$h:E(\eta)\rightarrow E((g|_{B(\eta)})^\ast \xi),$
$e\mapsto (\pi(e),g(e))$为同构.
笛卡尔积
给定向量丛$\xi_1,\xi_2,$ 定义$\xi_1\times \xi_2$为
$$
\pi_1\times\pi_2:E_1\times E_2\rightarrow B_1\times B_2,
$$
纤维为
$$
(\pi_1\times\pi_2)^{-1}(b_1,b_2)=F_{b_1}(\xi_1)\times F_{b_2}(\xi_2).
$$
Whitney和
对于同一底空间$B,$ 给定向量丛$\xi_1,\xi_2,$
对角映射$d:B\rightarrow B\times B.$
定义向量丛$\xi_1\oplus \xi_2:=d^\ast (\xi_1\times \xi_2),$ 其上纤维为
$$
F_b(d^\ast (\xi_1\times\xi_2))=F_b(\xi_1)\times F_b(\xi_2)\cong F_b(\xi_1)\oplus F_b(\xi_2).
$$
一般的, 向量空间上的操作都给出了向量丛上的操作, 依纤维操作即可,
如还可以定义向量丛上的张量积, 反称/对称化, 对偶从等.
欧氏向量丛
欧氏向量空间是一个向量空间,
配备了一个正定二次型$\mu:V\rightarrow \mathbb{R},$
即$\mu(\lambda v)=\lambda^2\mu(v),$ $\mu(v)>0,$ $\,\forall\,v\neq 0.$
那么$\mu$定义了$V$上的内积.
一个欧氏向量丛是向量丛, 每个纤维都是欧氏向量空间. 若$\xi$是欧氏的,
那么对于子丛$\eta,$ $\eta^\perp\oplus\eta=\xi.$
一个事实是每个Hausdorff, 预紧的底空间上的向量丛都可以给定欧氏度量.
CW复形, 度量空间, $\mathbb{R}^N$中流形上都可以给定欧氏度量.
文章最后更新于 2022-10-05 15:51:36
Basic concepts
for fiber bundles
$$
F\rightarrow E\rightarrow B,
$$
If $F$ is a $n$-dimensional (real) vector space,
And the local trivialization still maintains the vector space structure, it is called a rank $n$ (real) vector bundle. Tangent bundle, normal bundle,
Trivial vector bundles are all typical vector bundles.
There is a typical vector bundle on $\mathbb{R}\mathrm{P}^n$, which is a line bundle. It is defined as:
$$
E(\gamma_n^1):=\{(l,v)\in \mathbb{R}\mathrm{P}^n\times \mathbb{R}^{n+1}|v\in l\}.
$$
Let $\xi$ be the vector bundle $F\rightarrow E\xrightarrow{\pi}B.$
Definition $\xi$ on Section for mapping$s:B\rightarrow E,$
Let $B\xrightarrow{s}E\xrightarrow{\pi}B$ be the identity map.
The vector bundle $\xi\cong \eta$ on the same base space $B$ is called isomorphism,
If $\,\exists\,$ is homeomorphic $f:E(\xi)\rightarrow E(\eta),$
Make$\,\forall\,b\in B,$
$f|_{F_b(\xi)}:F_b(\xi)\cong F_b(\eta)$ is the isomorphism between vector spaces.
Proposition 1. $\,\forall\,n\ge 1,$ $\gamma_n^1$ has no non-trivial cross section that disappears everywhere. Therefore $\gamma_n^1$ is not isomorphic to the trivial bundle.
Let $s:\mathbb{R}\mathrm{P}^n\rightarrow E(\gamma_n^1)$ be a section, consider
$$
\mathbb{R}^{n+1}\setminus\{0\}\rightarrow \mathbb{R}\mathrm{P}^n\xrightarrow{s} E(\gamma_n^1),
$$
$$
v\mapsto ([v],t(v)v).
$$
However, $v,-v$ generates the same section, so $t(-v)=-t(v).$
Assume $t(v)>0,$ then $t(-v)<0,$
Then due to the connectivity of $\mathbb{R}^{n+1}\setminus \{0\}$, there must be a point $w$ such that $t(w)=0,$
Therefore, the cross section disappears at $[w]$. (Intermediate value theorem)
Proposition 2. A vector bundle $\xi$ of rank $n$ is trivial if and only if there are $n$ sections on $\xi$, which are linearly independent everywhere.
Find the trivial section of a trivial bundle. Suppose there are $n$ such sections,
Define mapping$B\times \mathbb{R}^n\xrightarrow{f} E(\xi),$
$$
\left(b,\sum t_ie_i\right)\mapsto \sum t_is_i(b)
$$
This is an isomorphism between vector bundles.
Construction of vector bundles
Induction bush (pull back bush)
Given a vector bundle $\xi,$ and a continuous map $B_1\rightarrow B,$, then
$$
f^\ast E:=\{(b,e)\in B_1\times E|f(b)=\pi(e)\}.
$$
That is, for $b,$
Pull back the fiber at $f(b)$ on $E$.
Define the bundle map between $\eta,\xi$ as a continuous map $E(\eta)\xrightarrow{g}E(\xi),$
So that it maintains the fiber and is linearly isomorphic on the fiber.
Lemma 3. Given the bundle map as above, we have $\eta\cong (g|_{B(\eta)})^\ast \xi.$
Just verify $h:E(\eta)\rightarrow E((g|_{B(\eta)})^\ast \xi),$
$e\mapsto (\pi(e),g(e))$ is isomorphism.
Cartesian product
Given the vector bundle $\xi_1,\xi_2,$, define $\xi_1\times \xi_2$ as
$$
\pi_1\times\pi_2:E_1\times E_2\rightarrow B_1\times B_2,
$$
The fiber is
$$
(\pi_1\times\pi_2)^{-1}(b_1,b_2)=F_{b_1}(\xi_1)\times F_{b_2}(\xi_2).
$$
Whitney and
For the same base space $B,$, the vector bundle $\xi_1,\xi_2,$ is given
Diagonal mapping$d:B\rightarrow B\times B.$
Define the vector bundle $\xi_1\oplus \xi_2:=d^\ast (\xi_1\times \xi_2),$ whose fibers are
$$
F_b(d^\ast (\xi_1\times\xi_2))=F_b(\xi_1)\times F_b(\xi_2)\cong F_b(\xi_1)\oplus F_b(\xi_2).
$$
Generally, operations on vector space provide operations on vector bundles, which can be done based on fiber operations.
For example, you can also define tensor products on vector bundles, inverse/symmetrization, duality, etc.
Euclidean vector bundle
Euclidean vector space is a vector space,
Equipped with a positive definite quadratic form $\mu:V\rightarrow \mathbb{R},$
That is $\mu(\lambda v)=\lambda^2\mu(v),$ $\mu(v)>0,$ $\,\forall\,v\neq 0.$
Then $\mu$ defines the inner product on $V$.
A Euclidean vector bundle is a vector bundle, and each fiber is a Euclidean vector space. If $\xi$ is Euclidean,
Then for the subcluster $\eta,$ $\eta^\perp\oplus\eta=\xi.$
It is a fact that for every Hausdorff, vector bundle on the pretensioned base space can be given a Euclidean metric.
Euclidean metrics can be given on CW complexes, metric spaces, and manifolds in $\mathbb{R}^N$.
The article was last updated on 2022-10-05 15:51:36
