# A question concerning the developing map of (G,X) manifolds

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 $$\varphi_U$$?

• 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? – YCor Feb 3 at 1:27
• @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. – 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.