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
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
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]