Rank of a vector bundle

Let $\pi:E\to M$ be a smooth real or complex vector bundle over a smooth manifold $M$. The rank of $E$ at a point $x\in M$ is

where $\mathbb F=\mathbb R$ for real bundles and $\mathbb F=\mathbb C$ for complex bundles (for example, a complex vector bundle ).

A smooth vector bundle is, by definition, locally trivial, so $\mathrm{rank}_x(E)$ is locally constant as a function of $x$. If $M$ is connected, then $\mathrm{rank}_x(E)$ is constant, and this common value is called the rank of $E$, denoted $\mathrm{rank}(E)$.

Examples

1. If $\dim M = n$, then $\mathrm{rank}(TM)=n$ and $\mathrm{rank}(T^*M)=n$ for the tangent bundle and cotangent bundle.

2. The trivial bundle $M\times \mathbb F^r\to M$ has rank $r$.

3. If $E$ and $F$ are bundles over the same base, then

   $$\mathrm{rank}(E\oplus F)=\mathrm{rank}(E)+\mathrm{rank}(F),\qquad \mathrm{rank}(E\otimes F)=\mathrm{rank}(E)\,\mathrm{rank}(F),$$

   where $\oplus$ and $\otimes$ denote the fiberwise direct sum and tensor product bundles.