Clifton-Pohl torus
The Clifton-Pohl torus is a counterexample to show that it is possible to have a geodesically incomplete compact semi-Riemannian manifold.
1. History
2. Topology
As the name implies, the Clifton-Pohl torus is topologically homeomorphic to a torus. It can be obtained from $\mathbb{R}^2 \setminus \{ 0 \}$ quotiented with some isometry $\Gamma$, as seen below.
3. Metrics and coordinates
The torus is constructed from the following metric on $M = \mathbb{R}^2 \setminus \{ 0 \}$
$$ds^2 = 2 \frac{dtdx}{t^2 + y^2}$$Homotheties are isometries of this metric :
$$(t,x) = (\lambda t, \lambda x)$$in particular the case $\lambda = 2$. If we consider the group $\Gamma$ generated by that transformation (homotheties by a factor of $2^n$, $\Gamma$ has a proper, discontinuous action on $M$.