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 $\varphi_U$?

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    $\begingroup$ It's hard to make sense of the question (the last sentence). What's "small enough $\tilde{U}$"? what's "for some $\varphi_U$" when $\varphi_U$ is already introduced? $\endgroup$ – YCor Feb 3 at 1:27
  • $\begingroup$ @YCor What I meant was for $\widetilde{U}$ diffeomorphic to both $U:=\mathfrak{p}(\widetilde{U})$ and $dev(\widetilde{U})$ , and $\varphi_U$ local coordinates on this open. $\endgroup$ – user135350 Feb 3 at 9:51

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.


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