Space form

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 $M_{K}$ .

This reduces the problem of studying space forms to studying discrete groups of isometries $\Gamma$ 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.

References

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