# Vector Bundles

Let $M$ be a differentiable manifold of dimension $n$. Consider an atlas $\mathcal{U}=\{U_\alpha\}_{\alpha\in\mathcal{A}}$ along with coordinates $x_\alpha^1,\cdots,x_\alpha^n$ in $U_\alpha$. For $x=(x_\alpha^1(x),\cdots,x_\alpha^n(x))\in U_\alpha$, a tangent vector is given by
$$v=\sum_{j=1}^nv_\alpha^j\frac{\partial}{\partial x_\alpha^j}.$$
If $x\in U_\alpha\cap U_\beta$, then $v$ is also written as
$$v=\sum_{j=1}^nv_\beta^j\frac{\partial}{\partial x_\beta^j}.$$
Here, the change of coordinates is given by
$$v_\beta^j=vx_\beta^j=\sum_{k=1}^nv_\alpha^k\frac{\partial x_\beta^j}{\partial x_\alpha^k}.$$
For $x\in U_\alpha\cap U_\beta$ and $f=(f^1,\cdots,f^n)\in\mathbb{R}^n$, define
\begin{align*}
h_{\alpha\beta}(x)(f)&=\left(\sum_{k=1}^n\frac{\partial x_\beta^1}{\partial x_\alpha^k}f^k,\cdots,(\sum_{k=1}^n\frac{\partial x_\beta^n}{\partial x_\alpha^k}f^k\right)\\
&=\begin{pmatrix}
\frac{\partial x_\beta^1}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^1}{\partial x_\alpha^n}\\
\vdots & \ddots & \vdots\\
\frac{\partial x_\beta^n}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^n}{\partial x_\alpha^n}
\end{pmatrix}\begin{pmatrix}
f^1\\
\vdots\\
f^n
\end{pmatrix}
\end{align*}
Hence $h_{\alpha\beta}:U_\alpha\cap U_\beta\longrightarrow\mathrm{Aut(\mathbb{R}^n)}$. The resulting bundle over $M$ with fibre $F=\mathbb{R}^n$ is called the tangent bundle of $M$ and is denoted by $TM$. Note that $TM$ is the set of all tangent vectors of $M$ i.e. $TM=\bigcup_{x\in M}T_xM$. For each $x\in U_\alpha$, the fibre $\pi^{-1}(x)$ of $x\in M$ is $T_xM\cong\{x\}\times\mathbb{R}^n$. The local trivialization map $h_\alpha:\pi^{-1}(U_\alpha)\longrightarrow U_\alpha\times\mathbb{R}^n$ is given by
$$h_\alpha(v)=(x,(v_\alpha^1,\cdots,v_\alpha^n)),\ v\in T_xU_\alpha(=T_xM),\ x\in U_\alpha.$$

A fibre bundle $(E,M,F,\pi)$ is called a vector bundle over $M$ if each fibre $F_x$ of $x\in M$ is a vector space. So, a tangent bundle is a vector bundle. The tangent bundle $TM$ is a differentiable manifold of dimension $2n$ with local coordinates in $\pi^{-1}(U_\alpha)$ being $(x_\alpha^1,\cdots,x_\alpha^n,v_\alpha^1,\cdots,v_\alpha^n)$. The Jacobian is given by
\begin{align*}
J(x_\beta^1,\cdots,x_\beta^n;x_\alpha^1,\cdots,x_\alpha^n)&=\frac{\partial(x_\beta^1,\cdots,x_\beta^n)}{\partial(x_\alpha^1,\cdots,x_\alpha^n)}\\
&=\begin{pmatrix}
\frac{\partial x_\beta^1}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^1}{\partial x_\alpha^n}\\
\vdots & \ddots & \vdots\\
\frac{\partial x_\beta^n}{\partial x_\alpha^1} & \cdots & \frac{\partial x_\beta^n}{\partial x_\alpha^n}
\end{pmatrix}:U_\alpha\cap U_\beta\longrightarrow\mathrm{GL}(n,\mathbb{R}).
\end{align*}
Let $g_{\alpha\beta}=J(x_\beta^1,\cdots,x_\beta^n;x_\alpha^1,\cdots,x_\alpha^n)$ Then $g_{\alpha\beta}$ satisfies
\begin{align*}
g_{\alpha\alpha}(x)&=I_n;\\
g_{\beta\alpha}(x)&=g_{\alpha\beta}^{-1}(x),\ x\in U_\alpha\cap U_\beta;\\
g_{\alpha\beta}(x)g_{\beta\gamma}(x)g_{\gamma\alpha}(x)&=I_n,\ x\in U_\alpha\cap U_\beta\cap U_\gamma.
\end{align*}
For $x\in U_\alpha\cap U_\beta$ and $f\in\mathbb{R}^n$, $h_{\alpha\beta}(x)(f)=g_{\alpha\beta}\cdot f$. So, $\mathrm{GL}(n,\mathbb{R})$ acts on the fibre $\mathbb{R}^n$. The map $g_{\alpha\beta}$ itself is often called a transition map.

If the transition map $h_\alpha\beta$ is the group action of a Lie group $G$ on the fibre $F$, the fibre bundle $(E,M,F,\pi)$ is called a $G$-bundle and the Lie group $G$ is called a structure group. The tangent bundle $TM$ is also a G-bundle with structure group $\mathrm{GL}(n,\mathbb{R})$.