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.

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