Fiber coordinates - Knowlpedia

Definition

Let $\Phi:\pi^{-1}(U)\to U\times F$ be a local trivialization of a fiber bundle. Writing

\Phi(p)=(\pi(p),f(p)),

the value $f(p)\in F$ , or ordinary coordinates on $F$ applied to it, is the fiber coordinate of $p$ in this trivialization.

Change of fiber coordinates

On the overlap of two trivializations, the base point remains fixed while the fiber coordinate changes by the associated transition function. Fiber coordinates therefore depend on a local trivialization and are not usually intrinsic to a point of the total space.

Principal bundles

For a principal $G$ -bundle, a local section chooses an origin in each fiber torsor and supplies a coordinate $h\in G$ . Changing the local section changes $h$ by multiplication with the principal-bundle transition function.

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: bundle coordinates and transition functions.