Geometric decompositions of surfaces with spherical metric and conical\n singularities

2020/07/07 by Guillaume Tahar, Tahar, Guillaume · 1 citation
Mathematics · #53A35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Mathematics and Applications #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2007.03355

openalex publication_date 2020/07/07 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We prove that any compact surface with constant positive curvature and\nconical singularities can be decomposed into irreducible components of standard\nshape, glued along geodesic arcs connecting conical singularities. This is a\nspherical analog of the geometric triangulations for flat surfaces with conical\nsingularities. The irreducible components include not only spherical triangles\nbut also other interesting spherical polygons. In particular, we present the\nclass of \half-spherical concave polygons that are spherical polygons\nwithout diagonals and that can be arbitrarily complicated. Finally, we\nintroduce the notion of core as a geometric invariant in the settings of\nspherical surfaces. We use it to prove a reducibily result for spherical\nsurfaces with a total conical angle at least (10g-10+5n)2\π.\n

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