fully defines the geometry of the space. For instance, if the Gaussian curvature vanishes throughout, the surface is Euclidean or at least flat.[32] In the same way, if the Gaussian curvature is an invariable positive or negative number throughout, the surface is one of constant curvature, either positive, as with a sphere, or negative, as with a pseudosphere.
But when we consider spaces of more than two dimensions, a knowledge of the generalised Gaussian curvature throughout the space is no longer sufficient to fix its geometry. While this curvature may vanish or present the same non-vanishing value throughout, we cannot infer therefrom that the space is necessarily flat or of constant curvature. Thus, whereas the vanishing of the generalised Gaussian curvature
is a necessary condition for the space to be flat, it is by no means sufficient. There is still room for a large measure of indeterminateness in the actual geometry of the space.
For the geometry of a space of more than two dimensions to be determined fully, we must state the values of the components of the tensor
(mentioned in the note); there being twenty such components at every point for a four-dimensional space. We may note that, contrary to the Gaussian curvature
,