A characterization of spherical polyhedron surfaces
  A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles.   For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obtained. A characterization of spherical polyhedron surfaces in terms of the triangulation and the angle assignment is established. 
Feng Luo