Definition:Local Trivialization

Definition
Let $\left( {E, M, \pi, F }\right)$ be a fiber bundle.

Let $\mathrm{pr_1} : M \times F \to M$ be the $1$st projection.

By definition, for every point $m \in M$ there exists an open neighborhood $U$ of $m$ and a homeomorphism:


 * $\displaystyle \chi : \pi^{-1} \left( { U } \right) \to U \times F$

such that:


 * $\displaystyle \pi {\restriction}_{ U}= \mathrm{pr}_{1} \circ \chi$

where $\displaystyle \pi {\restriction}_{ U}$ is the restriction of $\pi$ to $U$.

Then the ordered pair $\left( U, \chi \right)$ is called a local trivialization of $E$ over $U$.

Also See

 * Definition:Local Triviality


 * Definition:Transition Map