Local trivialization

A collection of local trivializations over an open cover that satisfy the compatibility condition defines a bundle atlas. On overlaps $U_i\cap U_j$, comparing trivializations produces the usual transition functions (or transition maps), and these satisfy the cocycle condition.

For principal bundles, one often uses the equivariant version (compare equivariant local trivialization), and local trivializations can be built from local sections as in constructing a trivialization from a local section.

For each $x\in U$, the bundle-compatible diffeomorphism $\Phi$ restricts to a diffeomorphism of fibers

1. Trivial bundle. For the product bundle $E=M\times F$ (see trivial fiber bundle), the global map
   $$\Phi:M\times F\to M\times F,\quad \Phi(x,f)=(x,f),$$
   is a local trivialization over every open $U\subset M$ (in fact a global trivialization).

1. Tangent bundle in a coordinate chart. Let $M$ be a smooth manifold and let $(U,\varphi)$ be a smooth chart (see smooth chart). The differential $d\varphi$ identifies $T_xM$ with $\mathbb R^n$ for $x\in U$. This yields a local trivialization of the tangent bundle over $U$,
   $$\Phi:TU\to U\times \mathbb R^n,\quad v_x\mapsto (x, d\varphi_x(v_x)).$$

1. Vector bundle via a local frame. If $E\to M$ is a rank-$n$ vector bundle and $(e_1,\dots,e_n)$ is a local frame on $U$, then every $v\in E_x$ can be written uniquely as $v=\sum_i a^i e_i(x)$. This gives a local trivialization
   $$\Phi:\pi^{-1}(U)\to U\times \mathbb R^n,\quad v\mapsto \big(\pi(v),(a^1,\dots,a^n)\big),$$
   and on overlaps the change of frame is encoded by the transition matrix.