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 with the metric
Any homothety is an isometry of , in particular including the map:
Let be the subgroup of the isometry group generated by . Then has a proper, discontinuous action on . Hence the quotient which is topologically the torus, is a Lorentz surface that is called the Clifton–Pohl torus. Sometimes, by extension, a surface is called a Clifton–Pohl torus if it is a finite covering of the quotient of by any homothety of ratio different from .
It can be verified that the curve
is a null geodesic that is incomplete. In fact, every null geodesic on or is incomplete.
The geodesic incompleteness of the Clifton–Pohl torus is better seen as a direct consequence of the fact that is extendable, i.e. that it can be seen as a subset of a bigger Lorentzian surface. It is a direct consequence of a simple change of coordinates. With
The metric (i.e. the metric expressed in the coordinates ) reads
But this metric extends naturally from to , where
The surface , known as the extended Clifton-Pohl plane, is geodesically complete.
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 i.e. "at infinity" in . Recall also that, by a theorem of E. Hopf no such tori exists in the Riemannian setting.
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- Bavard, Ch.; Mounoud, P. (2013), "Surfaces lorentziennes sans points conjugués", Geometry and Topology, 17: 469–492, doi:10.2140/gt.2013.17.469
- Hopf, E. (1948), "Closed surfaces without conjugate points", Proc. Natl. Acad. Sci. U.S.A., 34: 47–51, doi:10.1073/pnas.34.2.47