Consider some flat metric on $S^2$ with a fixed finite number of conical singularities $p\_1,\ldots,p\_n$.


What is the moduli space of such metrics up to isometry? In particular what is its dimension?



flat metrics on the 2-sphere with conical singularities
There is a necessary and sufficient condition that such a metric exist (at least when the cone angles are all less than $2\pi$), which is the simple linear condition on the cone angles imposed by Gauss-Bonnet. 


This allows for $n-1$ degrees of freedom. There is also an extra $2n$-dimensional freedom from moving the points independently of one another; however, one may use a conformal position to fix the positions of exactly three points, so there is really only $2n-6$ degrees of freedom obtained this way. Altogether the moduli space of flat metrics with cone angles less than $2\pi$ is $3n-7$ dimensional once one has modded out by the Möbius group. 


All of this (in the flat case particularly) follows immediately from work of Marc Troyanov, though certain global aspects of the moduli space require further arguments.


