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 . This reduces the problem of studying space forms to studying discrete groups of isometries 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 Borel conjecture References Riemannian geometry Conjectures