 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.The characteristic features of the geometry of non-positively curved Riemann surfaces are used to generalize the notion of non-positive beyond the study of Riemann surfaces.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.