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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 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.

- Goldberg, Samuel I. (1998),
*Curvature and Homology*, Dover Publications, ISBN 978-0-486-40207-9 - Lee, John M. (1997),
*Riemannian manifolds: an introduction to curvature*, Springer