Geometry of spheres with conical singularities

The classical Descartes’ theorem states that the total angular defect of a convex polyhedron is 4\pi. Such a polyhedron can be viewed as a flat surface homeomorphic to a sphere with cone singularities at vertices. We will explore the situation in both spherical geometry and hyperbolic geometry, use the Gauss-Bonnet theorem, and culminate the concept of stability.