Geometric angle structures on triangulated surfaces.pdf

a r X i v : m a t h / 0 6 0 1 4 8 6 v 1 [ m a t h .G T ] 2 0 J a n 2 0 0 6 Geometric angle structures on triangulated surfaces Ren Guo Abstract In this paper we characterize a function defined on the set of edges of a triangulated surface such that there is a spherical angle structure having the function as the edge invariant (or Delaunay invariant). We also characterize a function such that there is a hyperbolic angle structure having the function as the edge invariant. §1. Introduction Suppose S is a closed surface and T is a triangulation of S. Here by a triangulation we mean the following: take a finite collection of triangles and identify their edges in pairs by homeomorphism. Let V,E, F be the sets of all vertices, edges and triangles in T respectively. If a, b are two simplices in triangulation T , we use a b to denote that a is a face of b. Let C(S, T ) = {(e, f)|e ∈ E, f ∈ F, such that e f} be set of all corners of the triangulation. An angle structure on a triangulated surface (S, T ) assigns each corner of (S, T ) a number in (0, π). A Euclidean (or hyperbolic, or spherical) angles structure is an angle structure so that each triangle with the angle assignment is Euclidean (or hyperbolic, or spherical). More precisely, a Euclidean angle structure is a map x : C(S, T ) → (0, π) assigning every corner i (for simplicity of notation, we use one letter to denote a corner) a positive number xi such that xi+xj+xk = π whenever i, j, k are three corners of a triangle. A hyperbolic angle structure is a map x : C(S, T ) → (0, π) such that xi+xj+xk π. A spherical angle structure is a map x : C(S, T ) → (0, π) such that { xi + xj + xk π xj + xk ? xi π. (1) Actually it is proved in [B] that positive numbers xi, xj , xk are three inner angles of a spherical triangle if and only if they satisfy conditions (1). Given an angle structure x : C(S, T ) → (0, π), we define its edge invariant which is a function Dx : E → (0, 2π) such that Dx(e) = xi + xi′ where i =