Space form explained
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
with curvature
is isometric to,
hyperbolic space; with curvature
is isometric to,
Euclidean n-space; and with curvature
is isometric to
, the
n-dimensional sphere of points distance 1 from the origin in .
By rescaling the Riemannian metric on, we may create a space
of constant curvature
for any . Similarly, by rescaling the Riemannian metric on, we may create a space
of constant curvature
for any . Thus the universal cover of a space form
with constant curvature
is isometric to .
of
which act properly discontinuously. Note that the
fundamental group of,, will be isomorphic to . Groups acting in this manner on
are called
crystallographic groups. Groups acting in this manner on
and
are called
Fuchsian groups and
Kleinian groups, respectively.
See also
