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Reduction to generalized crystallographyEdit

The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form   with curvature   is isometric to  , hyperbolic space, with curvature   is isometric to  , Euclidean n-space, and with curvature   is isometric to  , the n-dimensional sphere of points distance 1 from the origin in  .

By rescaling the Riemannian metric on  , we may create a space   of constant curvature   for any  . Similarly, by rescaling the Riemannian metric on  , we may create a space   of constant curvature   for any  . Thus the universal cover of a space form   with constant curvature   is isometric to  .

This reduces the problem of studying space forms to studying discrete groups of isometries   of   which act properly discontinuously. Note that the fundamental group of  ,  , will be isomorphic to  . Groups acting in this manner on   are called crystallographic groups. Groups acting in this manner on   and   are called Fuchsian groups and Kleinian groups, respectively.

Space form problemEdit

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.

See alsoEdit

ReferencesEdit

  • 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