Space form

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 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 K < 0 . 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 form to studying discrete groups of isometries Γ of M K which act properly discontinuously. Note that the fundamental group of M , π 1 ( M ) , will be isomorphic to Γ . 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.

Space form problem

The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic fundamental groups are homeomorphic.

The possible extensions are limited. One might wish to conjecture that the manifolds are isometric, but rescaling the Riemannian metric on a compact aspherical Riemannian manifold preserves the fundamental group and shows this to be false. One might also wish to conjecture that the manifolds are diffeomorphic, but John Milnor's exotic spheres are all homeomorphic and hence have isomorphic fundamental group, showing this to be false.