is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements… - Howard P. Robertson

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 .

Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature <math>K = \frac{1}{R^2}</math>... [C]onsider... a "small circle" of radius <math>r</math> (measured on the surface!)... its perimeter <math>L</math> and area <math>A</math>... are clearly less than the corresponding measures <math>2\pi r</math> and <math>\pi r^2</math>... in the Euclidean plane. ...for sufficiently small <math>r</math> (i.e., small compared with <math>R</math>) these quantities on the sphere are given by 1):<math>L = 2 \pi r (1 - \frac{Kr^2}{6} + ...)</math>,
<math>A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)</math>

The solution of (1), which represents a homogeneous manifold, may be written in the form:<math>ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)</math>where <math>\kappa = \sqrt \frac{\lambda}{3}</math>. If we consider <math>\rho</math> as determining distance from the origin... and <math>\tau</math> as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world...