As I understand it, a real or complex vector bundle of rank $k$ consists of

- a manifold $X$ which is the base space of the vector bundle $E$
- a bundle projection $\pi: E \to X$
- for every $x \in X$ the \fibre $\pi^{-1}(x)\in F$ has the structure of a $R^n$- or $C^n$-space

Locally, $E$ is a product space in the following sense:

For every point $x\in X$ there is a suitable environment $U \subset X$ and $F_U\subset F$, a natural number $k$, and a mapping (a diffeomorphism) $\phi_U: U \times F_U \to \pi^{-1}(U)$ such that for all $x\in U$,

- $(\pi \circ \phi_U)(x,v) = x$ for all vectors $v \in F_U$
- The map $v \mapsto \phi_U(x,v)$ from $F_U$ to $\pi^{-1}(U)$

is linear and bijective.

The pair $(U,\phi_U)$ is called a local trivialization.

If the rank of the fibre $F$ is globally $k$, $E$ is a vector bundle of rank $k$}. If $k=1$ we have a

line bundle.

My question now is:

Do there exist any mathematical structures that look like a vector bundle for which the rank $k$ can not be defined globally, for example because it varies over the base space $X$?

And if such mathematical structures exist, what are they useful for in theoretical physics?