In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry.
may be endowed with a complete Riemannian metric with constant Gaussian curvature of either
As a result of the Gauss–Bonnet theorem one can determine that the surfaces which have a Riemannian metric of constant curvature
i.e. Riemann surfaces with a complete, Riemannian metric of non-positive constant curvature, are exactly those whose genus is at least
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.
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.
In the study of manifolds or orbifolds of higher dimension, the notion of sectional curvature is used wherein one restricts one's attention to two-dimensional subspaces of the tangent space at a given point.
In an arbitrary geodesic metric space the notions of being Gromov hyperbolic or of being a CAT(0) space generalise the notion that on a Riemann surface of non-positive curvature, triangles whose sides are geodesics appear thin whereas in settings of positive curvature they appear fat.