Let $B$ be a connected compact oriented topological surface, $B^{sing}\subset B$ a finite set. Assume that $B^{sm}=B\setminus B^{sing}$ carries a ${\bf Z}$-affine structure such that for any $x\in B^{sing}$ there exists a small neighborhood $U$ such that $U=\cup_{i\in I}U_{i}$ where $I$ is a finite set and each $U_{i}$ is affine equivalent to a germ of an angle in ${{\bf R}}^{2}$, with $x$ being the apex of each angle.

The aim of this section is to define a map $i_{loc}:B^{sing}\to{\frac{1}{12}}{{\bf Z}}$ (which depends only on ${\bf Z}$-affine structure near $B^{sing}$) and prove the following result (a kind of Gauss-Bonnet theorem).

We start with the construction of $i_{loc}$. Let us denote by $\widetilde{SL(2,{{\bf Z}})}$ the pre-image of $SL(2,{{\bf Z}})$ in the universal covering $\widetilde{SL(2,{{\bf R}})}$ of the group $SL(2,{{\bf R}})$. The group $\widetilde{SL(2,{{\bf R}})}$ contains $\pi_{1}(SL(2,{{\bf R}}))\simeq{{\bf Z}}$. Let $u$ be a generator of the latter (it belongs also to $\widetilde{SL(2,{{\bf Z}})}$).

We have an exact sequence of groups

Notice that $PSL(2,{{\bf Z}})$ is a free product ${{\bf Z}}/2*{{\bf Z}}/3$. Moreover, in the above exact sequence ${{\bf Z}}$ is embedded into the center of $\widetilde{SL(2,{{\bf Z}})}$. Notice that $u$ is the image of $2\in{{\bf Z}}$. One can choose representatives $a_{2},a_{3}$ of the standard generators of $PSL(2,{{\bf Z}})$ in such a way that $\widetilde{SL(2,{{\bf Z}})}$ is generated by $u,a_{2},a_{3}$ subject to the relations $a_{2}^{2}=a_{3}^{3},a_{2}^{4}=a_{3}^{6}=u$. This gives a homomorphism of groups $\phi:\widetilde{SL(2,{{\bf Z}})}\to{{\bf Z}}$ such that $\phi(a_{2})=3,\phi(a_{3})=2,\phi(u)=12$. Dividing by $12$ we obtain a homomorphism $i:\widetilde{SL(2,{{\bf Z}})}\to{\frac{1}{12}}{{\bf Z}}$ such that $i(u)=1$.

Let us consider a topological $S^{1}$-bundle $E$ over $B$, such that the fiber over $x\in B$ is the union of all affine rays outcoming of $x$. Then the restriction of $E$ to $B^{sm}$ is just the spherical bundle. Let us pick $x_{0}\in B^{sm}$ and remove it from $B$ together with small neighborhoods of all points $B^{sing}$. We denote by $B_{1}$ the topological space obtained in this way. We can trivialize the tangent bundle over $B_{1}$ (we choose a $C^{\infty}$-trivialization, compatible with $SL(2,{{\bf R}})$-structure) in such a way that it extends to a continuous trivialization of $S^{1}$-bundle $E$ over $B\setminus\{x_{0}\}$. Let $\alpha\in\Omega^{1}(B_{1})\otimes sl(2,{{\bf R}})$ be a $1$-form defined by means of the affine structure $\nabla$ on $B_{1}$. Then $d\alpha+{\frac{1}{2}}[\alpha,\alpha]=0$ and $\alpha$ defines a flat connection on a trivial $\widetilde{SL(2,{{\bf R}})}$-bundle on $B_{1}$. It gives a a monodromy representation $\pi_{1}(B_{1})\to\widetilde{SL(2,{{\bf Z}})}$ defined up to a conjugation. Composing it with the homomorphism $i$ we obtain a homomorphism $\pi_{1}(B_{1})\to{\frac{1}{12}}{{\bf Z}}$. Since ${\frac{1}{12}}{{\bf Z}}$ is an abelian group, the latter homomorphism is a composition $\pi_{1}(B_{1})\to H_{1}(B_{1},{\bf Z})\to{\frac{1}{12}}{{\bf Z}}$. Let us pick small circles $[\gamma_{x}]\in H_{1}(B,{\bf Z})$ for each $x\in B^{sing}$. Then the above homomorphism gives us a number denoted by $i_{loc}(x)\in{\frac{1}{12}}{{\bf Z}}$.

Proof of the Theorem. Let us pick up a small circle $[\gamma_{x_{0}}]\in H_{1}(B_{1},{\bf Z})$ around $x_{0}$. Then $\sum_{x\in B^{sing}}[\gamma_{x}]+[\gamma_{x_{0}}]=0$. The monodromy around $x_{0}$ can be easily computed via the winding number of the induced vector field (section of $E_{|\gamma_{x_{0}}}$) and is equal to $-\chi(B)u$. Applying homomorphism $i$ we obtain the result. $\blacksquare$

Proof. It is easy to see that for each point $x\in B^{sing}$ one has $i_{loc}(x)={\frac{1}{12}}$. Then from Gauss-Bonnet theorem one deduces that $\chi(B)=2-2g$, where $g$ is the genus of the Riemann surface $B$. Then we have

Since LHS is non-negative we conclude that either $g=1$ or $g=0$. In the first case $|B^{sing}|=0$ and we have a ${\bf Z}$-affine structure on a torus. In the second case we have $|B^{sing}|=24$ and $g=0$. $\blacksquare$

Similarly, for the affine structure with the monodromy at each point conjugate to

one has $i_{loc}(x)=\frac{1}{3}$, and we have six singular points on $S^{2}$ (see Section 4.2.5).