Space form
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In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K.
It is a theorem of Riemannian geometry that the universal cover of an n dimensional space form Mn with curvature K = − 1 is isometric to Hn, hyperbolic space, with curvature K = 0 is isometric to Rn, Euclidean n-space, and with curvature K = + 1 is isometric to Sn, the n-dimensional sphere of points distance 1 from the origin in Rn + 1.
By rescaling the Riemannian metric on Hn, we may create a space MK of constant curvature K for any K < 0. Similarly, by rescaling the Riemannian metric on Sn, we may create a space MK of constant curvature K for any K > 0. Thus the universal cover of a space form M with constant curvature K is isometric to MK.
This reduces the problem of studying space form to studying discrete groups of isometries Γ of MK which act properly discontinuously. Note that the fundamental group of M, π1(M), will be isomorphic to Γ. See the space form problem.
Groups acting in this manner on Rn are called crystallographic groups. Groups acting in this manner on H2 and H3 are called Fuchsian groups and Kleinian groups, respectively.
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