# Compact 3-manifolds of Negative Curvature

1. Aug 20, 2013

### StateOfTheEqn

Does anyone know if it is possible to construct a compact 3-manifold with no boundary and negative curvature? I ask this question in the Cosmology sub-forum because I see in various writings of cosmologists that it is often taken for granted that a negatively curved Universe must be infinite.

I think it can be proved, though I have not actually seen a proof, that it is impossible for 2-manifolds. Can anyone shed light on this question?

2. Aug 20, 2013

### marcus

http://www.jstor.org/discover/10.2307/1971158?uid=2129&uid=2&uid=70&uid=4&sid=21102540418401
Jorgensen (1977)
Compact 3-manifolds of constant negative curvature.

A more recent paper by John Milnor goes back over the history of classifying 3manifolds and on page 5 has a brief mention:
Although 3-manifolds of constant negative curvature actually exist in great variety, few examples were known until Thurston’s work in the late 1970’s. One interesting example was discovered already in 1912 by H. Gieseking. Starting with a regular 3-simplex of infinite edge length in hyperbolic 3-space, he identified the faces in pairs to obtain a non- orientable complete hyperbolic manifold of finite volume. Seifert and Weber described a compact example in 1933: Starting with a regular dodecahedron of carefully chosen size in hyperbolic space, they identified opposite faces by a translation followed by a rotation through 3/10-th of a full turn to obtain a compact orientable hyperbolic manifold. (An analogous construction using 1/10-th of a full turn yields Poincaré’s 3-manifold, with the 3-sphere as 120-fold covering space.)​
Here is an online version of Milnor's historical review:
http://www.math.sunysb.edu/~jack/PREPRINTS/tpc.pdf
Or see page 1229 of http://www.ams.org/notices/200310/fea-milnor.pdf

Last edited: Aug 20, 2013
3. Aug 21, 2013

### Chalnoth

You can clearly do this simply by identifying points on the manifold. A trivial example would be to take the FRW metric for negatively-curved space and identify points at a fixed distance from r=0 with the corresponding point in the opposite direction (that is, $\theta' = \pi - \theta, \phi' = \phi + \pi$, with $\phi$ being the azimuthal angle).