Geometric decompositions of surfaces with spherical metric and conical singularities

TL;DR

This paper establishes a method to decompose compact spherical surfaces with conical singularities into standard irreducible components, extending geometric triangulation concepts to spherical geometry and introducing a new invariant called the core.

Contribution

It introduces a decomposition framework for spherical surfaces with singularities, including new classes of polygons and the concept of the core as a geometric invariant.

Findings

Decomposition into irreducible spherical components including polygons.

Introduction of half-spherical concave polygons without diagonals.

Proof of reducibility for surfaces with large total conical angle.

Abstract

We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. This is a spherical analog of the geometric triangulations for flat surfaces with conical singularities. The irreducible components include not only spherical triangles but also other interesting spherical polygons. In particular, we present the class of \textit{half-spherical concave polygons} that are spherical polygons without diagonals and that can be arbitrarily complicated. Finally, we introduce the notion of core as a geometric invariant in the settings of spherical surfaces. We use it to prove a reducibily result for spherical surfaces with a total conical angle at least $(10g-10+5n)2\pi$.