Bundle atlas

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:

1. $\{U_i\}_{i\in I}$ is an open cover of $M$, and
2. 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

1. From a manifold atlas: the usual coordinate charts on $M$ induce a bundle atlas for $TM\to M$ by identifying $TM|_{U_i}\cong U_i\times\mathbb{R}^n$.
2. 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.
3. Principal bundles: local sections of a principal $G$-bundle yield local trivializations and hence a bundle atlas whose transitions take values in $G$.