Clifton-Pohl Torus
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Clifton%E2%80%93Pohl 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.[1] It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.[2]

Definition

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.[1] 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 .

Geodesic incompleteness

It can be verified that the curve

is a geodesic of M that is not complete (since it is not defined at ).[1] Consequently, (hence also ) is geodesically incomplete, despite the fact that is compact. Similarly, 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

consider

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.[3]

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.[3] The extended Clifton-Pohl plane contains 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.[4]

References

  1. ^ a b c O'Neill, Barrett (1983), Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, 103, Academic Press, p. 193, ISBN 9780080570570.
  2. ^ Wolf, Joseph A. (2011), Spaces of constant curvature (6th ed.), AMS Chelsea Publishing, Providence, RI, p. 95, ISBN 978-0-8218-5282-8, MR 2742530.
  3. ^ a b Bavard, Ch.; Mounoud, P. (2013), "Surfaces lorentziennes sans points conjugués", Geometry and Topology, 17: 469-492, doi:10.2140/gt.2013.17.469
  4. ^ Hopf, E. (1948), "Closed surfaces without conjugate points", Proc. Natl. Acad. Sci. U.S.A., 34: 47-51, Bibcode:1948PNAS...34...47H, doi:10.1073/pnas.34.2.47, PMC 1062913, PMID 16588785

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