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

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} .\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.

See also

• Borel conjecture