纤维丛

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 定义 9.1

一个（可微）纤维丛(fibre bundle) $(E,\pi ,M,F,G)$ 由下列要素组成：

(i) 一个可微流形 $E$，称为全空间(total space)。

(ii) 一个可微流形 $M$，称为底空间(base space)。

(iii) 一个可微流形 $F$，称为纤维(fibre)（或典型纤维）。

(iv) 一个满射 $\pi :E\to M$，称为投影映射(projection)。逆像 ${\pi }^{-1}(p)=F_p\cong F$ 称为点 $p$ 处的纤维。

(v) 一个李群 $G$，称为结构群(structure group)，它左作用在 $F$ 上。

(vi) 一组 $M$ 的开覆盖 $\{U_i\}$，以及微分同胚 ${\varphi }_i:U_i\times F\to {\pi }^{-1}(U_i)$，使得 $\pi \circ {\varphi }_i(p,f)=p$。映射 ${\varphi }_i$ 称为局部平凡化(local trivialization)，因为 ${\varphi }_i^{-1}$ 把 ${\pi }^{-1}(U_i)$ 映到直积 $U_i\times F$ 上。

(vii) 若记 ${\varphi }_i(p,f)={\varphi }_{i,p}(f)$，则映射 ${\varphi }_{i,p}:F\to F_p$ 是微分同胚。在 $U_i\cap U_j\ne \varnothing$ 上，我们要求 $t_{ij}(p)\equiv {\varphi }_{i,p}^{-1}\circ {\varphi }_{j,p}:F\to F$ 是 $G$ 的元素。于是 ${\varphi }_i$ 与 ${\varphi }_j$ 通过光滑映射 $t_{ij}:U_i\cap U_j\to G$ 相联系，如图 9.2所示

$${\varphi }_j(p,f)={\varphi }_i(p,t_{ij}(p)f).\text{\hspace{1em}(9.4)}$$

映射 $t_{ij}$ 称为过渡函数(transition function)。

[注：我们常用简写 $E\stackrel{\pi }{\to }M$ 或干脆用 $E$ 来表示纤维丛 $(E,\pi ,M,F,G)$。]

严格说来，纤维丛的定义应当与 $M$ 的特定覆盖 $\{U_i\}$ 无关。在数学文献中，这一说法被用来定义坐标丛 $(E,\pi ,M,F,G,\{U_i\},\{{\varphi }_i\})$。若 $(E,\pi ,M,F,G,\{U_i\}\cup \{V_j\},\{{\varphi }_i\}\cup \{{\psi }_j\})$ 仍是坐标丛，则称两个坐标丛 $(E,\pi ,M,F,G,\{U_i\},\{{\varphi }_i\})$ 与 $(E,\pi ,M,F,G,\{V_i\},\{{\psi }_i\})$ 等价。纤维丛定义为坐标丛的等价类。然而在物理学的实际应用中，我们总是采用某个

图 9.2. 在交叠区域 $U_i\cap U_j$ 上，同一元素 $u\in {\pi }^{-1}(p)$（其中 $p\in U_i\cap U_j$）被赋予两个元素 $f_i,f_j\in F$。它们通过 $t_{ij}(p)$ 相联系，如 $f_i=t_{ij}(p)f_j$。

确定的覆盖，并且不区分坐标丛与纤维丛。]

我们需要澄清几点。取底空间 $M$ 的一个坐标卡 $U_i$。${\pi }^{-1}(U_i)$ 是一个直积，与 $U_i\times F$ 微分同胚，其中 ${\varphi }_i^{-1}:{\pi }^{-1}(U_i)\to U_i\times F$ 就是该微分同胚。若 $U_i\cap U_j\ne \varnothing$，则在 $U_i\cap U_j$ 上有两个映射 ${\varphi }_i$ 和 ${\varphi }_j$。取一点 $u$，使得 $\pi (u)=p\in U_i\cap U_j$。于是我们得到 $F$ 的两个元素，一个由 ${\varphi }_i^{-1}$ 给出，另一个由 ${\varphi }_j^{-1}$ 给出，

$${\varphi }_i^{-1}(u)=\left(p,f_i\right),{\varphi }_j^{-1}(u)=\left(p,f_j\right)\text{\hspace{1em}(9.5)}$$

见图 9.2。存在映射 $t_{ij}:U_i\cap U_j\to G$，它把 $f_i$ 与 $f_j$ 联系起来，即 $f_i=t_{ij}(p)f_j$。这也写作(9.4)。

我们要求过渡函数满足如下相容性条件：

$$t_{ii}(p)=\text{恒等映射}(p\in U_i)\text{\hspace{1em}(9.6a)}$$ $$t_{ij}(p)=t_{ji}(p{)}^{-1}(p\in U_i\cap U_j)\text{\hspace{1em}(9.6b)}$$ $$t_{ij}(p)\cdot t_{jk}(p)=t_{ik}(p)(p\in U_i\cap U_j\cap U_k).\text{\hspace{1em}(9.6c)}$$

除非这些条件得到满足，纤维丛的各个局部片段就无法相容地粘合起来。若所有过渡函数都可取为恒等映射，则称该纤维丛为平凡丛(trivial bundle)。平凡丛就是直积 $M\times F$。

给定纤维丛 $E\stackrel{\pi }{\to }M$，可能的过渡函数集合显然远非唯一。设 $\{U_i\}$ 是 $M$ 的一个覆盖，$\{{\varphi }_i\}$ 与 $\{{\tilde{\varphi }}_i\}$ 是给出同一纤维丛的两组局部平凡化。各自的局部平凡化的过渡函数为

$$t_{ij}(p)={\varphi }_{i,p}^{-1}\circ {\varphi }_{j,p}\text{\hspace{1em}(9.7a)}$$ $${\tilde{t}}_{ij}(p)={\tilde{\varphi }}_{i,p}^{-1}\circ {\tilde{\varphi }}_{j,p}.\text{\hspace{1em}(9.7b)}$$

在每一点 $p\in M$ 处定义映射 $g_i(p):F\to F$ 为

$$g_i(p)\equiv {\varphi }_{i,p}^{-1}\circ {\tilde{\varphi }}_{i,p}.\text{\hspace{1em}(9.8)}$$

我们要求 $g_i(p)$ 是属于 $G$ 的同胚。若 $\{{\varphi }_i\}$ 与 $\{{\tilde{\varphi }}_i\}$ 描述的是同一个纤维丛，这一要求当然必须满足。由(9.7)和(9.8)容易看出

$${\tilde{t}}_{ij}(p)=g_i(p{)}^{-1}\circ t_{ij}(p)\circ g_j(p).\text{\hspace{1em}(9.9)}$$

在以后将要遇到的实际情况中，$t_{ij}$ 是把局部坐标卡粘合起来所需的规范变换，而 $g_i$ 则对应于坐标卡 $U_i$ 内部的规范自由度。若丛是平凡的，我们可以把所有过渡函数都取为恒等映射。此时过渡函数的最一般形式为

$$t_{ij}(p)=g_i(p{)}^{-1}{g}_j(p).\text{\hspace{1em}(9.10)}$$

设 $E\stackrel{\pi }{\to }M$ 是一个纤维丛。一个截面(section)（或横截面）$s:M\to E$ 是满足 $\pi \circ s={\mathrm{id}}_M$ 的光滑映射。显然，$s(p)=s{|}_p$ 是 $F_p={\pi }^{-1}(p)$ 的元素。$M$ 上截面的集合记作 $\Gamma (M,F)$。若 $U\subset M$，我们可以谈论只在 $U$ 上定义的局部截面(local section)。$\Gamma (U,F)$ 表示 $U$ 上局部截面的集合。例如，$\Gamma (M,TM)$ 等同于向量场的集合 $\mathcal{X}(M)$。应当注意，并非所有纤维丛都容许整体截面。

英文原文

A (differentiable) fibre bundle $(E,\pi ,M,F,G)$ consists of the following elements:

(i) A differentiable manifold $E$ called the total space.

(ii) A differentiable manifold $M$ called the base space.

(iii) A differentiable manifold $F$ called the fibre (or typical fibre).

(iv) A surjection $\pi :E\to M$ called the projection. The inverse image ${\pi }^{-1}(p)=F_p\cong F$ is called the fibre at $p$ .

(v) A Lie group $G$ called the structure group, which acts on $F$ on the left.

(vi) A set of open covering $\{U_i\}$ of $M$ with a diffeomorphism ${\varphi }_i:U_i\times F\to {\pi }^{-1}(U_i)$ such that $\pi \circ {\varphi }_i(p,f)=p$ . The map ${\varphi }_i$ is called the local trivialization since ${\varphi }_i^{-1}$ maps ${\pi }^{-1}(U_i)$ onto the direct product $U_i\times F$ .

(vii) If we write ${\varphi }_i(p,f)={\varphi }_{i,p}(f)$ , the map ${\varphi }_{i,p}:F\to F_p$ is a diffeomorphism. On $U_i\cap U_j\ne \varnothing$ , we require that $t_{ij}(p)\equiv {\varphi }_{i,p}^{-1}\circ {\varphi }_{j,p}:F\to F$ be an element of $G$ . Then ${\varphi }_i$ and ${\varphi }_j$ are related by a smooth map $t_{ij}:U_i\cap U_j\to G$ as (figure 9.2)

$${\varphi }_j(p,f)={\varphi }_i(p,t_{ij}(p)f).\text{\hspace{1em}(9.4)}$$

The maps $t_{ij}$ are called the transition functions.

[Remarks: We often use a shorthand notation $E\stackrel{\pi }{\to }M$ or simply $E$ to denote a fibre bundle $(E,\pi ,M,F,G)$ .]

Strictly speaking, the definition of a fibre bundle should be independent of the special covering $\{U_i\}$ of $M$ . In the mathematical literature, this definition is employed to define a coordinate bundle $(E,\pi ,M,F,G,\{U_i\},\{{\varphi }_i\})$ . Two coordinate bundles $(E,\pi ,M,F,G,\{U_i\},\{{\varphi }_i\})$ and $(E,\pi ,M,F,G,\{V_i\},\{{\psi }_i\})$ are said to be equivalent if $(E,\pi ,M,F,G,\{U_i\}\cup \{V_j\},\{{\varphi }_i\}\cup \{{\psi }_j\})$ is again a coordinate bundle. A fibre bundle is defined as an equivalence class of coordinate bundles. In practical applications in physics, however, we always employ a certain

Figure 9.2. On the overlap $U_i\cap U_j$ , two elements $f_i,f_j\in F$ are assigned to $u\in {\pi }^{-1}(p)$ , $p\in U_i\cap U_j$ . They are related by $t_{ij}(p)$ as $f_i=t_{ij}(p)f_j$ .

definite covering and make no distinction between a coordinate bundle and a fibre bundle.]

We need to clarify several points. Let us take a chart $U_i$ of the base space $M$ . ${\pi }^{-1}(U_i)$ is a direct product diffeomorphic to $U_i\times F$ , ${\varphi }_i^{-1}:{\pi }^{-1}(U_i)\to U_i\times F$ being the diffeomorphism. If $U_i\cap U_j\ne \varnothing$ , we have two maps ${\varphi }_i$ and ${\varphi }_j$ on $U_i\cap U_j$ . Let us take a point $u$ such that $\pi (u)=p\in U_i\cap U_j$ . We then assign two elements of $F$ , one by ${\varphi }_i^{-1}$ and the other by ${\varphi }_j^{-1}$ ,

$${\varphi }_i^{-1}(u)=\left(p,f_i\right),{\varphi }_j^{-1}(u)=\left(p,f_j\right)\text{\hspace{1em}(9.5)}$$

see figure 9.2. There exists a map $t_{ij}:U_i\cap U_j\to G$ which relates $f_i$ and $f_j$ as $f_i=t_{ij}(p)f_j$ . This is also written as (9.4).

We require that the transition functions satisfy the following consistency conditions:

$$t_{ii}(p)=\text{i d e n t i t y m a p}(p\in U_i)\text{\hspace{1em}(9.6a)}$$ $$t_{ij}(p)=t_{ji}(p{)}^{-1}(p\in U_i\cap U_j)\text{\hspace{1em}(9.6b)}$$ $$t_{ij}(p)\cdot t_{jk}(p)=t_{ik}(p)(p\in U_i\cap U_j\cap U_k).\text{\hspace{1em}(9.6c)}$$

Unless these conditions are satisfied, local pieces of a fibre bundle cannot be glued together consistently. If all the transition functions can be taken to be identity maps, the fibre bundle is called a trivial bundle. A trivial bundle is a direct product $M\times F$ .

Given a fibre bundle $E\stackrel{\pi }{\to }M$ , the possible set of transition functions is obviously far from unique. Let $\{U_i\}$ be a covering of $M$ and $\{{\varphi }_i\}$ and $\{{\tilde{\varphi }}_i\}$ be two sets of local trivializations giving rise to the same fibre bundle. The transition functions of respective local trivializations are

$$t_{ij}(p)={\varphi }_{i,p}^{-1}\circ {\varphi }_{j,p}\text{\hspace{1em}(9.7a)}$$ $${\tilde{t}}_{ij}(p)={\tilde{\varphi }}_{i,p}^{-1}\circ {\tilde{\varphi }}_{j,p}.\text{\hspace{1em}(9.7b)}$$

Define a map $g_i(p):F\to F$ at each point $p\in M$ by

$$g_i(p)\equiv {\varphi }_{i,p}^{-1}\circ {\tilde{\varphi }}_{i,p}.\text{\hspace{1em}(9.8)}$$

We require that $g_i(p)$ be a homeomorphism which belongs to $G$ . This requirement must certainly be fulfilled if $\{{\varphi }_i\}$ and $\{{\tilde{\varphi }}_i\}$ describe the same fibre bundle. It is easily seen from (9.7) and (9.8) that

$${\tilde{t}}_{ij}(p)=g_i(p{)}^{-1}\circ t_{ij}(p)\circ g_j(p).\text{\hspace{1em}(9.9)}$$

In the practical situations which we shall encounter later, $t_{ij}$ are the gauge transformations required for pasting local charts together, while $g_i$ corresponds to the gauge degrees of freedom within a chart $U_i$ . If the bundle is trivial, we may put all the transition functions to be identity maps. Then the most general form of the transition functions is

$$t_{ij}(p)=g_i(p{)}^{-1}{g}_j(p).\text{\hspace{1em}(9.10)}$$

Let $E\stackrel{\pi }{\to }M$ be a fibre bundle. A section (or a cross section) $s:M\to E$ is a smooth map which satisfies $\pi \circ s={\mathrm{id}}_M$ . Clearly, $s(p)=s{|}_p$ is an element of $F_p={\pi }^{-1}(p)$ . The set of sections on $M$ is denoted by $\Gamma (M,F)$ . If $U\subset M$ , we may talk of a local section which is defined only on $U$ . $\Gamma (U,F)$ denotes the set of local sections on $U$ . For example, $\Gamma (M,TM)$ is identified with the set of vector fields $\mathcal{X}(M)$ . It should be noted that not all fibre bundles admit global sections.

出处：9.2 纤维丛 · node a-a0ea8d42fdb5b94527b62728 · 源节点 SHA-256 769808919e7cffe6…