I'm having some trouble finding literature on the developing map.
All the sources I could find on it seem to refer to thurston's definition in either:
http://www.ucl.ac.uk/~ucahhjr/Notes/Essay.pdf
or
http://library.msri.org/books/gt3m/PDF/3.pdf
I don't have much background is geometric topology, so here I'm just trying to test my understanding:
(True or false?): Let $X = \mathbb{C}$ and $G$ be the pseudogroup of biholomorphisms between open subsets of $\mathbb{C}$. Then $\mathbb{C}$ with its usual complex structure is a $(G,X)$-manifold (ie, $(\text{biholo}(\mathbb{C}), \mathbb{C})$-manifolds are just Riemann surfaces). Furthermore, for any simply connected Riemann surface $M$, and any chart $\varphi : U\rightarrow\mathbb{C}$, where $U\subset M$, we have a map $D : M\rightarrow\mathbb{C}$ such that $D|_U = \varphi$.
(True or false?): Let $U\subseteq\mathbb{C}$ be a open disk around 1 on which a complex logarithm can be defined and is invertible. Then, we can view the topological space $\mathbb{C}$ as a Riemann surface via the atlas consisting of the two charts $\log : U\rightarrow\mathbb{C}$, and $\text{id} : \mathbb{C}\rightarrow\mathbb{C}$. This complex structure is equivalent to the usual complex structure on $\mathbb{C}$.
Question: What is the developing map from $\mathbb{C}\rightarrow\mathbb{C}$ relative to the chart $\log : U\rightarrow\mathbb{C}$?
I feel like this can't exist, but I don't see how thurston's definition of (G,X)-manifolds and the developing map manages to avoid its existence.
Another related weirdness is this: In the first link I gave, at the end of page 6, he describes a function $D : U_0\cup U_1\rightarrow X$ piecewise via the charts $\varphi_0 : U_0\rightarrow X,\;\;\varphi_1:U_1\rightarrow X$. However, his description doesn't even give a function on $U_0\cup U_1$, since the transition function $g$ isn't defined on $\varphi_1(U_1)$. It's only defined on $\varphi_1(U_0\cap U_1)$, and so his sending $x\mapsto g\circ\varphi_1(x)$ for all $x\in U_1$ doesn't make any sense. What's going on here?
thanks
Best Answer
False.
True.
It does not exist.
In his notes, before defining the developping map, Thurston specifies that $G$ has to act on the real analytic manifold $X$ by real analytic diffeomorphisms. In other words, he assumes that every $g \in G$ is defined everywhere on $X$. Hence a $(G,X)$-manifold is one such that its transition maps are restrictions of real analytic diffeomorphisms of $X$. That resolves the problem you point out in your last paragraph.
Since the $\log : U \to \mathbb{C}$ map is not the restriction of a global automorphism of $\mathbb{C}$, there is no associated developping map. More essentially, $\log$ cannot be analytically continued along any path going through the origin. The existence of a developping map depends on whether or not you can continue your chart analytically along every path in your manifold $M$. This is always possible if you restrict to groups $G$ as above.
When working with $\mathbb{C}$, the geometric manifolds which admit developping maps are affine surfaces, whose transition functions are affine maps. Among those are translation surfaces. If your model space is the unit disk $\mathbb{D}$ you get hyperbolic surfaces. All of these are special cases of projective (or Moebius) surfaces, modeled on the Riemann sphere $\widehat{\mathbb{C}}$.