Clifton–Pohl torus

In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete (by the Hopf–Rinow theorem), this space shows that the same implication does not generalize to pseudo-Riemannian manifolds. It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.

Consider the manifold ${\displaystyle \mathrm {M} =\mathbb {R} ^{2}\smallsetminus \{0\}}$ with the metric

Any homothety is an isometry of ${\displaystyle M}$, in particular including the map:

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