Spherical metrics with conical singularities on a 2-sphere: angle constraints

TL;DR

This paper establishes a precise criterion based on linear inequalities for the existence of spherical metrics with conical singularities and prescribed angles on a 2-sphere, under non-coaxial holonomy conditions.

Contribution

It provides a necessary and sufficient condition for such metrics to exist, formulated explicitly in terms of linear inequalities involving the cone angles.

Findings

Criterion expressed via linear inequalities for metric existence

Applicable to metrics with non-coaxial holonomy

Advances understanding of spherical metrics with singularities

Abstract

In this article we give a criterion for the existence of a metric of curvature $1$ on a $2$-sphere with $n$ conical singularities of prescribed angles $2\pi\vartheta_1,\dots,2\pi\vartheta_n$ and non-coaxial holonomy. Such a necessary and sufficient condition is expressed in terms of linear inequalities in $\vartheta_1,\dots,\vartheta_n$.