 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 formula_3) and the hyperbolic plane and pseudosphere (for curvature formula_5).