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 is isometric to , hyperbolic space is isometric to , Euclidean n-space, the n-dimensional sphere of points distance 1 from the origin in .

By rescaling the Riemannian metric on for any . Similarly, by rescaling the Riemannian metric on for any is isometric to .

This reduces the problem of studying space forms to studying discrete groups of isometries which act properly discontinuously. Note that the fundamental group of , , will be isomorphic to are called crystallographic groups are called Fuchsian groups and Kleinian groups, respectively.

See also

• Borel conjecture

References