9.1 Tangent bundles

For clarification, we begin our exposition with a motivating example. A tangent bundle $TM$ over an $m$ -dimensional manifold $M$ is a collection of all the tangent spaces of $M$ :

$$TM\equiv \bigcup _{p\in M}{T}_{p}M.\text{\hspace{1em}(9.1)}$$

The manifold $M$ over which $TM$ is defined is called the base space. Let $\{U_i\}$ be an open covering of $M$ . If $x^{\mu }={\phi }_i(p)$ is the coordinate on $U_i$ , an element of

$$T{U}_i\equiv \bigcup _{p\in U_i}{T}_{p}M$$

is specified by a point $p\in M$ and a vector $V=V^{\mu }(p)(\partial /\partial x^{\mu }){|}_p\in T_{p}M$ . Noting that $U_i$ is homeomorphic to an open subset $\phi (U_i)$ of ${\mathbb{R}}^m$ and each $T_{p}M$ is homeomorphic to ${\mathbb{R}}^m$ , we find that $T{U}_i$ is identified with a direct product ${\mathbb{R}}^m\times {\mathbb{R}}^m$ (figure 9.1). If $(p,V)\in T{U}_i$ , the identification is given by $(p,V)\mapsto (x^{\mu }(p),V^{\mu }(p))$ . $T{U}_i$ is a 2m-dimensional differentiable manifold. What is more, $T{U}_i$ is decomposed into a direct product $U_i\times {\mathbb{R}}^m$ . If we pick up a point $u$ of $T{U}_i$ , we can systematically decompose the information $u$ contains into a point $p\in M$ and a vector $V\in T_{p}M$ . Thus, we are naturally led to the concept of projection $\pi :T{U}_i\to U_i$ (figure 9.1). For any point $u\in T{U}_i$ , $\pi (u)$ is a point $p\in U_i$ at which the vector is defined. The information about the vector

Figure 9.1. A local piece $T{U}_i\simeq {\mathbb{R}}^m\times {\mathbb{R}}^m$ of a tangent bundle $TM$ . The projection $\pi$ projects a vector $V\in T_{p}M$ to $p$ .

is completely lost under the projection. Observe that ${\pi }^{-1}(p)=T_{p}M$ . In the context of the theory of fibre bundles, $T_{p}M$ is called the fibre at $p$ .

It is obvious by construction that if $M={\mathbb{R}}^m$ , the tangent bundle itself is expressed as a direct product ${\mathbb{R}}^m\times {\mathbb{R}}^m$ . However, this is not always the case and the non-trivial structure of the tangent bundle measures the topological nontriviality of $M$ . To see this, we have to look not only at a single chart $U_i$ but also at other charts. Let $U_j$ be a chart such that $U_i\cap U_j\ne \varnothing$ and let $y^{\mu }=\psi (p)$ be the coordinates on $U_j$ . Take a vector $V\in T_{p}M$ where $p\in U_i\cap U_j$ . $V$ has two coordinate presentations,

$$V=V^{\mu }\frac{\partial }{\partial x^{\mu }}{|}_p={\tilde{V}}^{\mu }\frac{\partial }{\partial y^{\mu }}{|}_p.\text{\hspace{1em}(9.2)}$$

It is easy to see that they are related as

$${\tilde{V}}^{\nu }=\frac{\partial y^{\nu }}{\partial x^{\mu }}(p)V^{\mu }.\text{\hspace{1em}(9.3)}$$

For $\{x^{\mu }\}$ and $\{y^{\nu }\}$ to be good coordinate systems, the matrix $(G_{\mu }^{\nu })\equiv (\partial y^{\nu }/\partial x^{\mu })$ must be non-singular: $(G_{\mu }^{\nu })\in \mathrm{GL}(m,\mathbb{R})$ . Thus, fibre coordinates are rotated by an element of $\mathrm{GL}(m,\mathbb{R})$ whenever we change the coordinates. The group $\mathrm{GL}(m,\mathbb{R})$ is called the structure group of $TM$ . In this way fibres are interwoven together to form a tangent bundle, which consequently may have quite a complicated topological structure.

We note en passant that the projection $\pi$ can be defined globally on $M$ . It is obvious that $\pi (u)=p$ does not depend on a special coordinate chosen. Thus, $\pi :TM\to M$ is defined globally with no reference to local charts.

Let $X\in \mathcal{X}(M)$ be a vector field on $M$ . $X$ assigns a vector $X{|}_p\in T_{p}M$ at each point $p\in M$ . From our viewpoint, $X$ is looked upon as a smooth map $M\to TM$ . This map is not utterly arbitrary since a point $p$ must be mapped to a point $u\in TM$ such that $\pi (u)=p$ . We define a section (or a cross section) of $TM$ as a smooth map $s:M\to TM$ such that $\pi \circ s={\mathrm{id}}_M$ . If a section $s_i:U_i\to T{U}_i$ is defined only on a chart $U_i$ , it is called a local section.

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF

来源版本：second-edition-local

来源 PDF SHA-256：fcd24c3371d092a8c699ccf5cf0765462924b6c48412804231076e8084b1940d

OCR 来源 SHA-256：0b6324f01172a3548cd34382e82fd8bfae574602b6b2cfaa357ba96bc2c35478

OCR 产物 SHA-256：2379d346cc6970295e7d6728588c4b68ec39a8957d07b95960aaae0f602300fe