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.

Contents

• Reduction to generalized crystallography
• See also
• References

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