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.
The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form
, hyperbolic space, with curvature
, Euclidean n-space, and with curvature
, the n-dimensional sphere of points distance 1 from the origin in
By rescaling the Riemannian metric on
of constant curvature
Similarly, by rescaling the Riemannian metric on
Thus the universal cover of a space form
This reduces the problem of studying space forms to studying discrete groups of isometries
which act properly discontinuously.
Note that the fundamental group of
Groups acting in this manner on
are called crystallographic groups.
Groups acting in this manner on
are called Fuchsian groups and Kleinian groups, respectively.