at
. Thus we see that whereas a two-dimensional surface has but one curvature at every point, a three-dimensional space has various curvatures at a point, depending on the direction in which the curvature is computed. If for all orientations of
, and for all points
, the curvature of the geodesic surface remains the same, we have a space of constant curvature. When the geodesic surfaces are spheres of the same radius, hence are surfaces of the same positive curvature, we have a spherical space; when pseudospheres, hence surfaces of constant negative curvature, we have a Lobatchewskian space, and when planes, hence surfaces of constant zero curvature, we have a Euclidean space.
It remains to be said that, as Riemann discovered, these curvatures of a space at a point can be described fully only in terms of the Riemann-Christoffel tensor
at each point. For this reason the Riemann-Christoffel tensor is the only one which enables us to define the nature of the space in a complete way.