Local trivialization

Let $\pi:E\to M$ be a smooth fiber bundle with typical fiber $F$ (see typical fiber ). Let $U\subset M$ be open.

A local trivialization of $E$ over $U$ is a diffeomorphism

such that the projection to $U$ agrees with $\pi$, i.e.

Equivalently, for each $x\in U$ the map $\Phi$ restricts to a diffeomorphism of fibers

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 .

Examples

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).

2. 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 $U$,

   $$\Phi:TU\to U\times \mathbb R^n,\quad v_x\mapsto (x, d\varphi_x(v_x)).$$

3. 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 .