Space form

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

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 forms to studying discrete groups of isometries $$\Gamma$$ of $$M_K$$ which act properly discontinuously. Note that the fundamental group of $$M$$, $$\pi_1(M)$$, will be isomorphic to $$\Gamma$$. 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.