Space form
In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.
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 Template:Tmath, hyperbolic space; with curvature <math>K = 0</math> is isometric to Template:Tmath, Euclidean n-space; and with curvature <math>K = +1</math> is isometric to <math>S^n</math>, the n-dimensional sphere of points distance 1 from the origin in Template:Tmath.
By rescaling the Riemannian metric on Template:Tmath, we may create a space <math>M_K</math> of constant curvature <math>K</math> for any Template:Tmath. Similarly, by rescaling the Riemannian metric on Template:Tmath, we may create a space <math>M_K</math> of constant curvature <math>K</math> for any Template:Tmath. Thus the universal cover of a space form <math>M</math> with constant curvature <math>K</math> is isometric to Template:Tmath.
This reduces the problem of studying space forms to studying discrete groups of isometries <math>\Gamma</math> of <math>M_K</math> which act properly discontinuously. Note that the fundamental group of Template:Tmath, Template:Tmath, will be isomorphic to Template:Tmath. Groups acting in this manner on <math>R^n</math> are called crystallographic groups. Groups acting in this manner on <math>H^2</math> and <math>H^3</math> are called Fuchsian groups and Kleinian groups, respectively.