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I was reading Wikipedia about $(G,X)$-manifolds, and I do not understand well the developing map part.

https://en.wikipedia.org/wiki/(G,X)-manifold#Developing_map

Here is what Wikipedia says. Let $M$ be a connected $(G,X)$-manifold, and let $\pi : \tilde{M} \to M$ be the universal covering. A developing map $\varphi : \tilde{M} \to X$ is defined as follows.

  1. Fix $p \in \tilde{M}$. Let $q \in \tilde{M}$ be given.

  2. Fix a chart $\varphi : U \to X$ near $p$. Consider a path $\gamma$ from $p$ to $q$.

  3. We may use analytic continuation along $\gamma$ to extend $\varphi$ so that its domain includes $q$ (maybe $\gamma$?).

  4. Since $\tilde{M}$ is simply connected, $\varphi(q)$ does not depend on the choice of $\gamma$. (The monodromy theorem ensures the well-definedness of $\varphi : \tilde{M} \to X$.)

At the step 3, why does the analytic continuation exist? How do we extend $\varphi$ along $\gamma$?

Thank you.

D.L
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1 Answers1

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This is because of the "analytic continuation" property of $(G,X)$-manifolds (that is, locally a $(G,X)$-manifold is a real analytic manifold $X$ carrying an action of the Lie group $G$ by real analytic diffeomorphisms; see e.g. Version of identity theorem for functions in $\mathbb{R}^n$ or Identity theorem for $\mathbb{R}^n$ or The Identity Theorem for real analytic functions for analytic continuation) together with the fact that any cover of a $(G,X)$-manifold (in particular the universal cover) has a unique $(G,X)$-manifold structure that makes the covering map a $(G,X)$-map (see e.g. Exr.5.1.1 on p.109 of Goldman's Geometric Structures on Manifolds (https://www.math.umd.edu/~wmg/gstom.pdf)): cover the path $\gamma$ by finitely many charts; they necessarily intersect and so $\varphi$ can be extended to a real analytic map defined on their union.

Also see Prop.5.2.1 on p.112 of Goldman's notes for a detailed argument for 4..

Alp Uzman
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