Bundle atlas - Knowlpedia

Let $\pi:E\to M$ be a smooth fiber bundle with typical fiber $F$ . A bundle atlas is a collection of local trivializations $\{(U_i,\Phi_i)\}_{i\in I}$ such that:

$\{U_i\}_{i\in I}$ is an open cover of $M$ , and
for every overlap $U_{ij}:=U_i\cap U_j$ , the change of trivialization $\Phi_i\circ \Phi_j^{-1}:U_{ij}\times F\longrightarrow U_{ij}\times F$ is a diffeomorphism over $\mathrm{id}_{U_{ij}}$ , hence is of the form $(x,f)\mapsto (x, t_{ij}(x)(f))$ for a smooth family of diffeomorphisms of $F$ .

The associated maps $t_{ij}$ are the transition functions of the atlas and satisfy the standard cocycle identities on triple overlaps.

Examples

From a manifold atlas: the usual coordinate charts on $M$ induce a bundle atlas for $TM\to M$ by identifying $TM|_{U_i}\cong U_i\times\mathbb{R}^n$ .
Möbius line bundle: two local trivializations over overlapping arcs of $S^1$ form a bundle atlas; the overlap map is given by a sign change in the fiber.
Principal bundles: local sections of a principal $G$ -bundle yield local trivializations and hence a bundle atlas whose transitions take values in $G$ .