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 $M^n$ with curvature $K=-1$ is isometric to Template:Tmath, hyperbolic space; with curvature $K=0$ is isometric to Template:Tmath, 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 Template:Tmath.

By rescaling the Riemannian metric on Template:Tmath, we may create a space $M_K$ of constant curvature $K$ for any Template:Tmath. Similarly, by rescaling the Riemannian metric on Template:Tmath, we may create a space $M_K$ of constant curvature $K$ for any Template:Tmath. Thus the universal cover of a space form $M$ with constant curvature $K$ is isometric to Template:Tmath.

This reduces the problem of studying space forms to studying discrete groups of isometries $\mathrm{\Gamma }$ of $M_K$ 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 $R^n$ are called crystallographic groups. Groups acting in this manner on $H^2$ and $H^3$ are called Fuchsian groups and Kleinian groups, respectively.

See also

• Borel conjecture