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Space form

## Summary

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 ${\displaystyle M^{n}}$  with curvature ${\displaystyle K=-1}$  is isometric to ${\displaystyle H^{n}}$ , hyperbolic space, with curvature ${\displaystyle K=0}$  is isometric to ${\displaystyle R^{n}}$ , Euclidean n-space, and with curvature ${\displaystyle K=+1}$  is isometric to ${\displaystyle S^{n}}$ , the n-dimensional sphere of points distance 1 from the origin in ${\displaystyle R^{n+1}}$ .

By rescaling the Riemannian metric on ${\displaystyle H^{n}}$ , we may create a space ${\displaystyle M_{K}}$  of constant curvature ${\displaystyle K}$  for any ${\displaystyle K<0}$ . Similarly, by rescaling the Riemannian metric on ${\displaystyle S^{n}}$ , we may create a space ${\displaystyle M_{K}}$  of constant curvature ${\displaystyle K}$  for any ${\displaystyle K>0}$ . Thus the universal cover of a space form ${\displaystyle M}$  with constant curvature ${\displaystyle K}$  is isometric to ${\displaystyle M_{K}}$ .

This reduces the problem of studying space forms to studying discrete groups of isometries ${\displaystyle \Gamma }$  of ${\displaystyle M_{K}}$  which act properly discontinuously. Note that the fundamental group of ${\displaystyle M}$ , ${\displaystyle \pi _{1}(M)}$ , will be isomorphic to ${\displaystyle \Gamma }$ . Groups acting in this manner on ${\displaystyle R^{n}}$  are called crystallographic groups. Groups acting in this manner on ${\displaystyle H^{2}}$  and ${\displaystyle H^{3}}$  are called Fuchsian groups and Kleinian groups, respectively.