- Space form
In
mathematics , a space form is a completeRiemannian manifold "M" of constantsectional curvature "K".Reduction to generalized crystallography
It is a theorem of Riemannian geometry 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 form 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 calledcrystallographic group s. Groups acting in this manner on and are calledFuchsian group s andKleinian group s, respectively.pace form problem
The space form problem is a conjecture stating that any two compact aspherical Riemannian manifolds with isomorphic
fundamental group s 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, butJohn Milnor 'sexotic sphere s are all homeomorphic and hence have isomorphic fundamental group, showing this to be false.ee also
*
Borel conjecture
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