Complex vector bundle

Let $M$ be a smooth manifold . A (smooth) complex vector bundle over $M$ is a smooth surjective map $\pi:E\to M$ together with the following data and properties:

1. For each $x\in M$, the fiber $E_x:=\pi^{-1}(x)$ is a finite-dimensional complex vector space.

2. There is an open cover $\{U_\alpha\}$ of $M$ and smooth maps (local trivializations)

   $$\Phi_\alpha:\pi^{-1}(U_\alpha)\to U_\alpha\times \mathbb C^r$$

   such that:

   • $\mathrm{pr}_1\circ \Phi_\alpha=\pi$ on $\pi^{-1}(U_\alpha)$, and
   • for each $x\in U_\alpha$, the induced map $(\Phi_\alpha)_x:E_x\to \{x\}\times\mathbb C^r\cong \mathbb C^r$ is complex linear.

Equivalently, on overlaps $U_\alpha\cap U_\beta$ the transition maps

have the form $(x,v)\mapsto (x,g_{\alpha\beta}(x)v)$ for a smooth map $g_{\alpha\beta}:U_\alpha\cap U_\beta\to \mathrm{GL}(r,\mathbb C)$.

The integer $r$ is the (complex) rank; it is locally constant and hence constant if $M$ is connected (see rank of a vector bundle ).

Examples

1. Trivial bundle. For any $r\ge 1$, the projection $M\times\mathbb C^r\to M$ is a complex vector bundle with the obvious trivializations.

2. Complexified tangent and cotangent bundles. The bundles $TM\otimes_\mathbb R \mathbb C$ and $T^*M\otimes_\mathbb R \mathbb C$ are complex vector bundles obtained by complexifying the tangent bundle and cotangent bundle .

3. Bundle of complex-valued $k$-forms. The exterior power $\Lambda^k(T^*M\otimes\mathbb C)$ is a complex vector bundle whose smooth sections are complex-valued differential k-forms .