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

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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

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It can be verified that the curve

 

is a null 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 also 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

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The Clifton–Pohl tori are also remarkable by the fact that they were the first known non-flat Lorentzian tori with no conjugate points.[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 Hopf–Rinow theorem no such tori exists in the Riemannian setting.[4]

References

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  1. ^ a b c O'Neill, Barrett (1983), Semi-Riemannian Geometry With Applications to Relativity, Pure and Applied Mathematics, vol. 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 (2): 47–51, Bibcode:1948PNAS...34...47H, doi:10.1073/pnas.34.2.47, PMC 1062913, PMID 16588785