Space form

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 . 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 $\backslash Gamma$ of $M\_K$ which act properly discontinuously. Note that the fundamental group of $M$, $\backslash pi\_1(M)$, will be isomorphic to $\backslash Gamma$. 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.

• Borel conjecture

• {{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