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.
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Definition
Consider the manifold
Any homothety is an isometry of
Let
Geodesic incompleteness
It can be verified that the curve
is a geodesic of M that is not complete (since it is not defined at
is a null geodesic that is incomplete. In fact, every null geodesic on
The geodesic incompleteness of the Clifton–Pohl torus is better seen as a direct consequence of the fact that
consider
The metric
But this metric extends naturally from
The surface
Conjugate points
The Clifton–Pohl tori are also remarkable by the fact that they are the only non-flat Lorentzian tori with no conjugate points that are known. It is interesting to note that the extended Clifton–Pohl plane does contain a lot of pairs of conjugate points, some of them being in the boundary of