Let $M$ be a $(G,X)$ manifold, that is we have local charts $(U,\varphi\_U)$ on $M$ with $\varphi\_U$ a diffeomorphism onto an open subset of $X$ and the transition maps are locally-$G$.


Let $\mathfrak{p}:\widetilde{M}\rightarrow M$ be the universal covering of $M$.


The *developing map theorem* introduces a local diffeomorphism $dev:\widetilde{M}\rightarrow X$.


Does the developing map locally commute with the (restricted) covering and local charts, i.e. for $\widetilde{U}$ (small enough) $dev|\_\widetilde{U}=\varphi\_{U}\circ\mathfrak{p}|\_\widetilde{U}$ for some $\varphi\_U$?



A question concerning the developing map of (G,X) manifolds
The best way to see this is to read the proof of the existence of the developing map.
The $(X,G)$ structure on $M$ induces an $(X,G)$-structure on $\tilde M$ its universal cover such that for any $\tilde x\in \tilde M$, write $x=p(\tilde x)$ where $p$ is the covering map. Consider a chart $x\in U$ such that there exists $\tilde U$ which contains $\tilde x$ such that the restriction of $p$ to $\tilde U$, $p\_{\tilde U}:\tilde U\rightarrow U$ is an homeomorphism $\phi\_{\tilde U}=\phi\_U\circ p\_{\tilde U}$.


To construct the developing map, one fixes $\tilde x\_0\in \tilde M$ for every $\tilde x\in \hat M$, one considers a path $c:[0,1]\rightarrow \hat M$ such that $c(0)=\tilde x\_0$ and $c(1)=\tilde x$, then one consider chart $(\tilde U\_0,\tilde \phi\_0),...(\tilde U\_n,\tilde\phi\_n)$ such that there exists a subdivision $[0=t\_0,t\_1,...,t\_n=1]$ such that $c[t\_i,t\_{i+1}]\subset \tilde U\_i$ and one sets:


$D(\tilde x)=\tilde g\_0...g\_{n-1}\tilde\phi\_n(\tilde x)$ where $g\_i\in G$ such that $\tilde \phi\_i\circ{\tilde\phi\_{i+1}}^{-1}$ is the restriction of $g\_i$ to $\tilde\phi\_{i+1}(\tilde U\_{i+1})$.


This shows that if we set $U\_n=p(\tilde U\_n)$, we can define the chart $(U\_n,g\_0...g\_{n-1}\phi\_n)$ of $x$ and the chart $(\tilde U\_n,g\_0...g\_{n-1}\tilde \phi\_n)$ which answer your question.


