Space form

In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious 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 ${\displaystyle K=-1}$ is isometric to $H^{n}$, hyperbolic space, with curvature ${\displaystyle K=0}$ is isometric to $R^{n}$, Euclidean n-space, and with curvature ${\displaystyle K=+1}$ is isometric to $S^{n}$, the n-dimensional sphere of points distance 1 from the origin in ${\displaystyle R^{n+1}}$.

By rescaling the Riemannian metric on $H^{n}$, we may create a space ${\displaystyle M_{K}}$ of constant curvature $K$ for any ${\displaystyle K<0}$. Similarly, by rescaling the Riemannian metric on $S^{n}$, we may create a space ${\displaystyle M_{K}}$ of constant curvature $K$ for any $K>0$. Thus the universal cover of a space form $M$ with constant curvature $K$ is isometric to ${\displaystyle M_{K}}$.

This reduces the problem of studying space form to studying discrete groups of isometries $\Gamma$ of ${\displaystyle M_{K}}$ 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.

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.

See also

• Borel conjecture

References

• 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