Editing Space form (section)

==Reduction to generalized crystallography==

The  [[Killing–Hopf theorem]] of Riemannian geometry states that the [[universal cover]] of an ''n''-dimensional space form <math>M^n</math> with curvature <math>K = -1</math> is isometric to <math>H^n</math>, [[hyperbolic space]], with curvature <math>K = 0</math> is isometric to <math>R^n</math>, [[Euclidean space|Euclidean ''n''-space]], and with curvature <math>K = +1</math> is isometric to <math>S^n</math>, the [[N-sphere|n-dimensional sphere]] of points distance 1 from the origin in <math>R^{n+1}</math>.

By rescaling the [[Riemannian metric]] on <math>H^n</math>, we may create a space <math>M_K</math> of constant curvature <math>K</math> for any <math>K < 0</math>.  Similarly, by rescaling the Riemannian metric on <math>S^n</math>, we may create a space <math>M_K</math> of constant curvature <math>K</math> for any <math>K > 0</math>. Thus the universal cover of a space form <math>M</math> with constant curvature <math>K</math> is isometric to <math>M_K</math>.

This reduces the problem of studying space forms to studying [[discrete space|discrete]] [[group (mathematics)|groups]] of [[isometry|isometries]] <math>\Gamma</math> of <math>M_K</math> which act [[properly discontinuous]]ly. Note that the [[fundamental group]] of <math>M</math>, <math>\pi_1(M)</math>, will be isomorphic to <math>\Gamma</math>. Groups acting in this manner on <math>R^n</math> are called [[crystallographic group]]s. Groups acting in this manner on <math>H^2</math> and <math>H^3</math> are called [[Fuchsian group]]s and [[Kleinian group]]s, respectively.