If S is a closed, orientable
Riemann surface then it follows from the
Uniformization theorem that S may be endowed with a complete
Riemannian metric with constant
Gaussian curvature of either 0, 1 or -1. As a result of the
Gauss–Bonnet theorem one can determine that the surfaces which have a Riemannian metric of constant curvature 0 -1 i.e. Riemann surfaces with a complete, Riemannian metric of non-positive constant curvature, are exactly those whose
genus is at least 1. 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 1. The definition of curvature above depends upon the existence of a
Riemannian metric and therefore lies in the field of geometry. However the Gauss–Bonnet theorem ensures that the topology of a surface places constraints on the complete Riemannian metrics which may be imposed on a surface so the study of metric spaces of non-positive curvature is of vital interest in both the mathematical fields of
geometry and
topology. Classical examples of surfaces of non-positive curvature are the
Euclidean plane and
flat torus (for curvature 0) and the
hyperbolic plane and
pseudosphere (for curvature -1). For this reason these metrics as well as the Riemann surfaces which on which they lie as complete metrics are referred to as Euclidean and hyperbolic respectively. == Generalizations ==