Space form

---

title: "Space form" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["riemannian-geometry", "conjectures"] topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Space_form" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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 , hyperbolic space; with curvature K = 0 is isometric to , 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 .

By rescaling the Riemannian metric on , we may create a space M_K of constant curvature K for any {{tmath| K M_K of constant curvature K for any . Thus the universal cover of a space form M with constant curvature K is isometric to .

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 , , will be isomorphic to . 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

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::