Geometric decompositions of spherical surfaces

Abstract

It is well-known that any plane polygon can be decomposed along its diagonals into flat triangles. The analog does not hold for spherical polygons. 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. The irreducible components include not only spherical triangles but also other interesting spherical polygons called half-spherical concave polygons.

As an application, we prove that any spherical surface with a total angle at least (10g−10+5n)2π contains an embedded sphere with a slit.