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?
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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.
answered Oct 20, 2022 at 13:32
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$\begingroup$ I can't see where in the paper you reference Troyanov actually discusses the case of one conical point. Or were you suggesting to deduce it from the case of two conical points? Something like, a sphere with one conical point can be viewed as a sphere with two conical points, one having conical angle $2\pi$. Then, by Troyanov's result, the other must have conical angle $2\pi$? $\endgroup$ Commented Oct 20, 2022 at 14:00
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$\begingroup$ Yes, you understand correctly. Just introduce the second singularity with angle $2\pi$ and apply Troyanov's result, if you are not satisfied with the proof I gave. $\endgroup$ Commented Oct 20, 2022 at 14:05
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$\begingroup$ The proof you gave is perfectly satisfactory; however, I was hoping to be able to simply quote a result (which I can now also do, thanks to your further explanation!) $\endgroup$ Commented Oct 20, 2022 at 14:16
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$\begingroup$ In the papers about metrics with conic singularities, the case of 1 singularity is considered trivial, and it is usually only briefly mentioned that such metrics on the sphere do not exist. (In the paper that I cited it is not even mentioned). There is a complete description of possible sets of angles at the conic singularities for metrics on the sphere, MR3556430, MR4105912. The statement is of course a formal consequence of the results of these papers. $\endgroup$ Commented Oct 20, 2022 at 20:46