9.1 Tangent Bundles

A tangent bundle $TM$ over an $m$ -dimensional manifold $M$ is a collection of all tangent spaces of $M$ :

$\begin{equation} TM\equiv \bigcup_{p\in M} T_{p}M \end{equation}$

The manifold $M$ over which $TM$ is defined is called base space.

Let $\{U_{i}\}$ be an open covering of $M$ , and $x^{\mu}=\varphi_{i}(p)$ is the coordinate on $U_{i}$ , an element of $TU_{i}\equiv \bigcup_{p\in U_{i}} T_{p}M$ is specified by a point $p\in M$ and a vector $V\in T_{p}M$ . We find that $TU_{i}$ is identified with $\mathbb{R}^{m}\times \mathbb{R}^{m}$ . If $(p,V)\in T U_{i}$ , the identification is given by $(p,V)\mapsto (x^{\mu}(p),V^{\mu}(p))$ , $TU_{i}$ is a $2m$ -dim differentiable manifold. We can define a natural projection $\pi:TU_{i}\to U_{i}$ , $\pi(p,V)=p$ , and $\pi ^{-1}(p)=T_{p}M$ . In the context of fibre bundle theory, $T_{p}M$ is called a fibre at $p$ .

If $M=\mathbb{R}^{m}$ then $TM=\mathbb{R}^{m}\times \mathbb{R}^{m}$ , but it is not the case for general $M$ , and $TM$ measures topological nontriviality of $M$ . To see this we need to look at different charts $U_{i},U_{j}$ , with $U_{i}\cap U_{j}\neq \emptyset$ , let $x^{\mu},y^{\mu}$ be their coordinates respectively. For $V\in T_{p}M,p\in U_{i}\cap U_{j}$ ,

$\begin{equation} V= V^{\mu} \frac{ \partial }{ \partial x^{\mu} }|_{p} = \tilde{V}^{\mu} \frac{ \partial }{ \partial y^{\mu} }|_{p} \end{equation}$

and $\tilde{V}^{\mu}=\frac{ \partial y^{\nu} }{ \partial x ^{\mu}}(p)V^{\mu}\equiv G^{\nu}_{\mu}V^{\mu}$ . $G\in GL(m,\mathbb{R})$ must be non-singular. The fibre coordinates are rotated whenever we switch to another coordinates. $GL(m,\mathbb{R})$ is the structure group of $TM$ .

9.1 Tangent Bundles

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