Clifton-Pohl Torus

Get Clifton%E2%80%93Pohl Torus essential facts below. View Videos or join the Clifton%E2%80%93Pohl Torus discussion. Add Clifton%E2%80%93Pohl Torus to your PopFlock.com topic list for future reference or share this resource on social media.
## Definition

## Geodesic incompleteness

## Conjugate points

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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]}

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 .

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]}

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]}

- ^
^{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. **^**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.- ^
^{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 **^**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

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists