ABSTRACT
Every Riemannian space in which the curvature is bounded above by a number K is a space of curvature not greater than K . However, a space of curvature not greater than K is generally neither a Riemannian space nor a manifold. For instance, the figure formed by two plane (Euclidean) triangles meeting a t a common vertex is a two-dimensional space of non positive curvature (provided the distance between two points is defined as the length of a shortest arc joining the points in this figure).
