Note 34 — Chapter VII · SYSTEMS OF CO-ORDINATES AND DISTANCE

We may note that the geometry, hence the nature, of a space is fully determined only when we express it in terms of the Riemann-Christoffel tensor . Thus at every point denotes perfectly flat or homaloidal space; , on the other hand, comprises a whole series of generally non-homogeneous spaces of which (the homogeneous and homaloidal variety) is only one particular illustration. Hence, by itself does not yield us any very definite information.

Likewise, a perfectly homogeneous spherical space of four dimensions, i.e., the type of space which corresponds to Riemann’s geometry of four dimensions, is given by where is a constant depending on the intensity of the curvature. On the other hand, represents a whole series of spaces, generally non-homogeneous, of which the above-mentioned truly spherical and homogeneous variety is only one particular case.