Clifton–Pohl 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 $\mathrm{M} = \mathbb{R}^2 \smallsetminus \{0\}$ with the metric

$g= \frac{2 dx dy}{x^2+y^2}$

Any homothety is an isometry of $M$, in particular including the map:

$\lambda(x,y)=(2x,2y)$

Let $\Gamma$ be the subgroup of the isometry group generated by $\lambda$. Then $\Gamma$ has a proper, discontinuous action on $M$. Hence the quotient $T = M/\Gamma,$ 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 $M$ by any homothety of ratio different from $\pm 1$.

Geodesic incompleteness

It can be verified that the curve

$\sigma(t) = \left(\frac{1}{1-t},0\right)$

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

$\sigma(t) = (\tan t, 1)$

is a null geodesic that is incomplete. In fact, every null geodesic on $M$ or $T$ is incomplete.

The geodesic incompleteness of the Clifton–Pohl torus is better seen as a direct consequence of the fact that $(M,g)$ 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. Consider

$\begin{align} F\colon ]-\pi/2,\pi/2[^2\smallsetminus\{0\} \ &\to M \\ (u,v) &\mapsto (\tan(u),\tan(v)). \end{align}$

The metric $F^*g$ (i.e. the metric $g$ expressed in the coordinates $(u,v)$) reads

$\widehat g=\frac{2 du dv}{\cos^2 u\sin^2 v+\sin ^2 u \cos^2 v}.$

But this metric extends naturally from $]-\pi/2,\pi/2[^2\smallsetminus\{0\}$ to $\mathbb R^2 \smallsetminus \Lambda$, where

$\Lambda =\{\frac{\pi}{2}(k,l);\, (k,l)\in \mathbb Z^2, k+l\equiv 0(\text{mod}\,2)\}.$

The surface $(\mathbb R^2 \smallsetminus \Lambda, \widehat g)$, 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] 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 $]-\pi/2,\pi/2[^2$ i.e. "at infinity" in $M$. Recall also that, by a theorem of E. Hopf no such tori exists in the Riemannian setting.^[4]

References

1. 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. Bavard, Ch.; Mounoud, P. (2013), "Surfaces lorentziennes sans points conjugués", Geometry and Topology 17: 469–492
4. Hopf, E. (1948), "Closed surfaces without conjugate points", Proc. Nat. Acad. Sci. U. S. A. 34: 47–51