Vector bundle trivialization
= Vector bundle trivialization
A local trivialization of a rank-$r$ <vector bundle> $\pi:E\to M$ is a bundle isomorphism $\pi^{-1}(U)\cong U\times\mathbb R^r$ over an open set $U$, linear on each fiber. Overlapping trivializations differ by $(p,v)\mapsto(p,g(p)v)$ for a smooth $GL_r(\mathbb R)$-valued function $g$. A <tangent bundle> chart uses the derivative of its base <manifold chart> for this fiber identification.