Clifton–Pohl torus

Consider the manifold \mathrm{M} = \mathbb{R}^2 \smallsetminus \{0\} with the metric : g= \frac{2\,dx\,dy}{x^2+y^2}=\frac{dx\otimes dy + dy\otimes dx}{x^2+y^2} Any homothety is an isometry of , in particular including the map: : \lambda(x,y)=(2x, 2y) Let \Gamma be the subgroup of the isometry group generated by . Then \Gamma 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 M by any homothety of ratio different from . == Geodesic incompleteness ==

It can be verified that the curve : \sigma(t) := \left(\frac 1 {1-t},0\right) is a null geodesic of M that is not complete (since it is not defined at ). == 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. == References ==