For a smooth principal $G$-bundle $P\to M$ with local product charts over an open cover $(U_i)_{i\in I}$, the transition maps $g_{ij}:U_i\cap U_j\to G$ satisfy

\begin{align*} g_{ii}(p)=e \end{align*}

and

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

for all $p\in U_i\cap U_j\cap U_k$. Conversely, if $(U_i)_{i\in I}$ is an open cover of the smooth manifold $M$ and smooth maps $g_{ij}:U_i\cap U_j\to G$ satisfy these identities on every nonempty double and triple overlap, then the quotient of $\bigsqcup_i(U_i\times G)$ by the [equivalence relation](/page/Equivalence%20Relation) generated, for $p\in U_i\cap U_j$, by

\begin{align*} (p,h)_j\sim (p,g_{ij}(p)h)_i \end{align*}

admits a unique smooth structure for which the quotient maps $U_i\times G\to P$ are inverse local trivialization charts, equivalently whose inverses are local trivializations $\pi^{-1}(U_i)\to U_i\times G$, and it determines a smooth principal $G$-bundle up to isomorphism.