is not invariant. It splits up into components each one of which may vary in value when the mesh-system is changed. Yet, in spite of this variability,
still defines the curvatures of the space, hence still refers to something that is intrinsic and irrelevant to our choice of mesh-system. Such is one of the characteristics of a tensor. This particular tensor of twenty components (in a four-dimensional space) is known as the Riemann-Christoffel tensor.[33] Curiously enough, it was discovered by Riemann, not when investigating the geometry of space, but when considering a problem in heat.
Only when every one of these twenty components of the tensor
vanishes at every point is it possible to assert that the four-dimensional space is Euclidean, or at least flat. Thus, whereas for two dimensions Euclideanism was ensured when one condition was satisfied at every point, namely, the vanishing of the Gaussian curvature,
, on the other hand, when we step up to four dimensions, twenty conditions are needed, and these are given by the vanishing at every point of the twenty expressions which in their aggregate constitute the Riemann-Christoffel tensor
.