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?



1 Answer 1




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 $\widehat{\mathbb{C}}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.