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.

See also

• Borel conjecture

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