Steiner Ratio for Manifolds
  The Steiner ratio characterizes the greatest possible deviation of the length of a minimal spanning tree from the length of the minimal Steiner tree. In this paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As a corollary, the Steiner ratio for flat tori, flat Klein bottles, and projective plane of constant positive curvature are computed. Steiner ratio - Steiner problem - Gilbert--Pollack conjecture - surfaces of constant curvature 
D. Cieslik, A.O. Ivanov, A.A. Tuzhilin
math.MG