 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. It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.

## Definition

Consider the manifold $\mathrm {M} =\mathbb {R} ^{2}\smallsetminus \{0\}$ with the metric

$g={\frac {dx\,dy}{{\tfrac {1}{2}}(x^{2}+y^{2})}}$ Any homothety is an isometry of $M$ , in particular including the map:

$\lambda (x,y)=2\cdot (x,y)$ 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. 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 $t=1$ ). 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. With

$N=\left(-\pi /2,\pi /2\right)^{2}\smallsetminus \{0\};$ consider

$F:N\to M$ $F(u,v):=(\tan(u),\tan(v)).$ The metric $F^{*}g$ (i.e. the metric $g$ expressed in the coordinates $(u,v)$ ) reads

${\widehat {g\,}}={\frac {du\,dv}{{\tfrac {1}{2}}(\cos(u)^{2}\sin(v)^{2}+\sin(u)^{2}\cos(v)^{2})}}.$ But this metric extends naturally from $N$ to $\mathbb {R} ^{2}\smallsetminus \Lambda$ , where

$\Lambda =\left\{{\tfrac {\pi }{2}}(k,\ell )\ \mid \ (k,\ell )\in \mathbb {Z} ^{2},k+\ell \equiv 0{\pmod {2}}\right\}.$ The surface $(\mathbb {R} ^{2}\smallsetminus \Lambda ,{\widehat {g\,}})$ , known as the extended Clifton-Pohl plane, is geodesically complete.

## 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. The extended Clifton-Pohl plane contains 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.