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 M^n with curvature K = -1 is isometric to H^n, hyperbolic space, with curvature K = 0 is isometric to R^n, Euclidean n-space, and with curvature K = +1 is isometric to S^n, the n-dimensional sphere of points distance 1 from the origin in R^{n+1}. By rescaling the Riemannian metric on H^n, we may create a space M_K of constant curvature K for any K < 0. Similarly, by rescaling the Riemannian metric on S^n, we may create a space M_K of constant curvature K for any K > 0. Thus the universal cover of a space form M with constant curvature K is isometric to M_K. This reduces the problem of studying space forms to studying discrete groups of isometries \Gamma of M_K which act properly discontinuously. Note that the fundamental group of M, \pi_1(M), will be isomorphic to \Gamma. Groups acting in this manner on R^n are called crystallographic groups. Groups acting in this manner on H^2 and H^3 are called Fuchsian groups and Kleinian groups, respectively.