Space form

 ( Riemannian Geometry ) 

In mathematics, a space form is a complete space Riemannian manifold M of constant sectional curvature K. The three most fundamental examples are Euclidean space, the n-sphere, and hyperbolic space, although a space form need not be simply connected.

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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 , hyperbolic space; with curvature $K\; =\; 0$ is isometric to , Euclidean space; and with curvature $K\; =\; +1$ is isometric to $S^n$, the N-sphere of points distance 1 from the origin in .

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

This reduces the problem of studying space forms to studying discrete space groups of isometry $\backslash Gamma$ of $M\_K$ which act properly discontinuously. Note that the fundamental group of , , will be isomorphic to . 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 and , respectively.

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See also

• Borel conjecture

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