This... is an outrageously over-simplified account of the assumptions and procedures... - Howard P. Robertson

The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature <math>K</math> provides formulae for the surface area <math>S</math> and the volume <math>V</math> of a "small sphere" of radius <math>r</math>, whose leading terms are 3)<math>S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...)</math>,
<math>V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)</math>.

The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients <math>g_{\mu \nu}</math> satisfy the differential equations<math>G_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)</math>in all regions free from matter and electromagnetic field, where <math>G_{\mu \nu}</math> is the contracted Riemann-Christoffel tensor associated with the fundamental tensor <math>g_{\mu \nu}</math>, and <math>\lambda</math> is the .

The field equation may... be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium—in complete analogy with... the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish.