Let \(M\) be a topological space. A vector fiber bundle of rank \(k\) over \(M\) is a topological space \(E\) together with a surjective map \(\pi\colon E\to M\) satisfying,
- for all \(p \in M\) the fiber \(E_p=\pi^{-1}(p)\) is a \(k\)-dimensional vector space
- for all \(p \in M\) exists a neighbourhood \(U\subset M\) of \(p\) and a homeomorphism \(\rho\colon U\times \mathbb{R}^k\to \pi^{-1}(U)\) (called local trivialization) such that
- for each \(p\in U\) \(\rho(p,\cdot )\) is a vector space isomorphism form \(\{p\}\times \mathbb{R}^k\) to \(E_p\) and
- \(\pi\circ \rho(p,\cdot )=p\).
We denote the vector fiber section bundle by the tuple \((E,\pi,M)\).