Definition of a line bundle and examples

The simplest example of a line bundle over a manifold $M$ is the trivial bundle $\mathbb{C}\times M$. Here the vector space at each point $m$ is $\mathbb{C}\times \{ m\}$ which we regard as a copy of $\mathbb{C}$. The general definition uses this as a local model.

Definition 1.1   A complex line bundle over a manifold $M$ is a manifold $L$ and a smooth surjection $\pi \colon L \to M$ such that:

1.

Each fibre $\pi^{-1} (m) = L_m$ is a a complex one-dimensional vector space.

2.

Every $m \in M$ has an open neighbourhood $U \in M$ for which there is a diffeomeorphism $\varphi : \pi^{-1} (U) \to U \times \mathbb{C}$ such that $\varphi (L_m) \subset \{ m \} \times \mathbb{C}$ for every $m$ and that moreover the map $\varphi\vert _{L_{m}} : L_{m} \to \{m\} \times \mathbb{C}$ is a linear isomorphism.

Note 1.1   The second condition is called local triviality because it says that locally the line bundle looks like $\mathbb{C}\times M$. We leave it as an exercise to show that local triviality makes the map $\pi$ a submersion (that is it has onto derivative) and the scalar multiplication and vector addition maps smooth. In the quantum mechanical example local triviality means that at least in some local region like the laboratory we can identify the Hermitian vector space where the wave function takes its values with $\mathbb{C}$.

Example 1.1   $\mathbb{C}\times M$ the trivial bundle

Example 1.2   Recall that if $u \in S^2$ then the tangent space at $u$ to $S^2$ is identified with the set $T_{u} S^{2} = \{ v \in \mathbb{R}^{3} \mid \< v , u \> = 0 \}$. We make this two dimensional real vector space a one dimensional complex vector space by defining $(\alpha + i \beta) v = \alpha. v + \beta. u \times v.$ We leave it as an exercise for the reader to show that this does indeed make $T_uS^2$ into a complex vector space. What needs to be checked is that $[ (\alpha + i \beta) \ ( \delta + i \gamma)] v = (\alpha + i \beta) \ [ (\delta + i \gamma)] v$ and because $T_uS^2$ is already a real vector space this follows if $i(iv) = -v$. Geometrically this follows from the fact that we have defined multiplication by $i$ to mean rotation through $\pi/2$. We will prove local triviality in a moment.

Example 1.3   If $\Sigma$ is any surface in $\mathbb{R}^3$ we can use the same construction as in (2). If $x \in \Sigma$ and $\hat{n}_{x}$ is the unit normal then $T_{x} \Sigma = \hat{n}_{x}^{\perp}$. We make this a complex space by defining $(\alpha + i \beta) v = \alpha v + \beta \hat{n}_{x} \times v$.

Example 1.4 (Hopf bundle)   Define $\mathbb{C}P_{1}$ to be the set of all lines (through the origin) in $\mathbb{C}^{2}$. Denote the line through the vector $z = (z^{0}, z^{1})$ by $[z] = [z^{0}, z^{1}]$. Note that $[\lambda z^{0}, \lambda z^{1}] = [z^{0}, z^{1}]$ for any non-zero complex number $\lambda$. Define two open sets $U_{i}$ by

$$U_{i} = \{ [z^{0}, z^{1}] \mid z^{i} = 0\}$$

and co-ordinates by $\psi_{i} \colon U_{i} \to \mathbb{C}$ by $\psi_{0}([z]) = z^{1}/z^{0}$ and $\psi_{1}([z]) = z^{0}/z^{1}$. As a manifold $\mathbb{C}P_{1}$ is diffeomorphic to $S^2$. An explicit diffeomorphism $S^{2} \to \mathbb{C}P_{1}$ is given by $(x, y, z) \mapsto [x+ i y, 1-z]$.

We define a line bundle $H$ over $\mathbb{C}P_{1}$ by $H \subset \mathbb{C}^{2} \times \mathbb{C}P_{1}$ where

$$H = \{ (w, [z]) \mid \hbox{$w = \lambda z$ for some $ \lambda \in \mathbb{C}^{\times}$}\}.$$

We define a projection $\pi \colon H \to \mathbb{C}P_{1}$ by $\pi((w, [z])) = [z]$. The fibre $H_[z] = \pi^{-1}([z])$ is the set

$$\{ (\lambda z, [z]) \mid \lambda \in \mathbb{C}^{\times} \}$$

which is naturally identified with the line through $[z]$. It thereby inherits a vector space structure given by

$$\alpha(w, [z]) + \beta (w', [z]) = (\alpha w + \beta w', [z]).$$

We shall prove later that this is locally trivial.