  In his study of curved surfaces, Gauss introduced a real-valued function, the Gaussian curvature, which measures a surface's local deviation from flatness in terms of the surface's intrinsic geometry. Riemann extended this concept of curvature to Riemannian n-manifolds. By using his extended concept of curvature, he was able to characterize with great elegance the metric manifolds in which all figures can freely move around without changing their size and shape. They are the Riemannian manifolds of constant curvature. This idea can be nicely combined with Klein's classification of metric geometries. Regarded as Riemannian 3-manifolds, Euclidean space has constant zero curvature, Lobachevskian space has constant negative curvature, and elliptic space has constant positive curvature. Pursuant to the Erlangen Program, each of these geometries of constant curvature is characterized by its own group of isometries. But Klein's conception is too narrow to embrace all Riemannian geometries, which include spaces of variable curvature. Indeed, in the general case, the group of isometries of a Riemannian n-manifold is the trivial group consisting of the identity alone, whose structure conveys no information at all about the respective geometry.