Clifton–Pohl torus

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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

Multiplication by any real number 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.[1]

Geodesic incompleteness

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.


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