Let $M$ be a smooth manifold, let $(U_i)_{i \in I}$ be an open cover of $M$, let $F$ be a smooth manifold, and let

\begin{align*} g_{ij}: U_i \cap U_j \to \operatorname{Diff}(F) \end{align*}

be transition functions such that the induced maps

\begin{align*} G_{ij}: (U_i\cap U_j)\times F\to (U_i\cap U_j)\times F \end{align*}

defined by $G_{ij}(x,v)=(x,g_{ij}(x)v)$ are smooth diffeomorphisms and satisfying

\begin{align*} g_{ii}(x)=\operatorname{id}_F. \end{align*}

They also satisfy

\begin{align*} g_{ij}(x)g_{jk}(x)=g_{ik}(x) \end{align*}

for all $x \in U_i \cap U_j \cap U_k$. Let $E$ be the quotient set of $\bigsqcup_i U_i \times F$ by

\begin{align*} (x,v,i) \sim (x,g_{ji}(x)v,j) \end{align*}

and define $\pi:E\to M$ by sending the class of $(x,v,i)$ to $x$. There is a unique topology and smooth structure on $E$ for which the maps induced by the summands $U_i\times F$ are local trivializations. Assume that this atlas topology is Hausdorff and second-countable. Then $\pi:E\to M$ is a smooth fibre bundle with fibre $F$. If $F = \mathbb R^k$ and each $g_{ij}$ takes values in $GL(k,\mathbb R)$, the resulting bundle is a rank-$k$ smooth vector bundle.