It seems to be considered a classical fact that one cannot have a spherical polyhedral/cone-metric on the 2-sphere with precisely one conical point. However, I've never actually seen it proven anywhere in full generality. I realise that it's not too hard to prove using the holonomy and developing map, but I would prefer a reference for my purposes. Does anyone know of a reference?



Nonexistence of sphere with one conical point [reference request]
The proof is very simple. Let $f$ be the developing map (take an isometry of some small disk on your surface to a region in the plane with constant curvature metric, and then perform analytic continuation along all paths not passing through singularity). The monodromy representation is in the group of isometries of your plane. Since the sphere minus one point is simply connected, this monodromy is trivial. Trivial monodromy means that your map is a ramified covering. But there is no ramified covering from the sphere to anything, ramified only at one point.


The reference is
MR1034288
Troyanov, Marc
Metrics of constant curvature on a sphere with two conical singularities. Differential geometry (Peñíscola, 1988), 296–306,
Lecture Notes in Math., 1410, Springer, Berlin, 1989.


