Clifton–Pohl torus

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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[edit]

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[edit]

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[edit]

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] 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.[4]

References[edit]

  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, doi:10.1073/pnas.34.2.47