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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. 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 ). 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
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.
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 i.e. "at infinity" in .
Recall also that, by a theorem of E. Hopf no such tori exists in the Riemannian setting.