 Riemann surfaces with a complete, Riemannian metric of non-positive constant curvature, are exactly those whose genus is at least formula_4.The Uniformization theorem and the Gauss–Bonnet theorem can both be applied to orientable Riemann surfaces with boundary to show that those surfaces which have a non-positive Euler characteristic are exactly those which admit a Riemannian metric of non-positive curvature.There is therefore an infinite family of homeomorphism types of such surfaces whereas the Riemann sphere is the only closed, orientable Riemann surface of constant Gaussian curvature formula_4.