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 <math>M^n</math> with curvature <math>K = -1</math> is isometric to {{tmath| H^n }}, hyperbolic space; with curvature <math>K = 0</math> is isometric to {{tmath| R^n }}, Euclidean ''n''-space; and with curvature <math>K = +1</math> is isometric to <math>S^n</math>, the ''n''-dimensional sphere of points distance 1 from the origin in {{tmath| R^{n+1} }}.
By rescaling the Riemannian metric on {{tmath| H^n }}, we may create a space <math>M_K</math> of constant curvature <math>K</math> for any {{tmath| K < 0 }}. Similarly, by rescaling the Riemannian metric on {{tmath| S^n }}, we may create a space <math>M_K</math> of constant curvature <math>K</math> for any {{tmath| K > 0 }}. Thus the universal cover of a space form <math>M</math> with constant curvature <math>K</math> is isometric to {{tmath| M_K }}.
This reduces the problem of studying space forms to studying discrete groups of isometries <math>\Gamma</math> of <math>M_K</math> which act properly discontinuously. Note that the fundamental group of {{tmath| M }}, {{tmath| \pi_1(M) }}, will be isomorphic to {{tmath| \Gamma }}. Groups acting in this manner on <math>R^n</math> are called crystallographic groups. Groups acting in this manner on <math>H^2</math> and <math>H^3</math> are called Fuchsian groups and Kleinian groups, respectively.
== See also == * Borel conjecture
== References == * {{citation | last1=Goldberg | first1=Samuel I. | title=Curvature and Homology | publisher=Dover Publications | isbn=978-0-486-40207-9 | year=1998 }} * {{citation | last1=Lee | first1=John M. | title=Riemannian manifolds: an introduction to curvature | publisher=Springer | year=1997 }}
Category:Riemannian geometry Category:Conjectures