Clifton–Pohl torus

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

${\displaystyle g={\frac {dx\,dy}{{\tfrac {1}{2}}(x^{2}+y^{2})}}}$

Any homothety is an isometry of ${\displaystyle M}$, in particular including the map:

${\displaystyle \lambda (x,y)=(2x,2y)}$

Let ${\displaystyle \Gamma }$ be the subgroup of the isometry group generated by ${\displaystyle \lambda }$. Then ${\displaystyle \Gamma }$ has a proper, discontinuous action on ${\displaystyle M}$. Hence the quotient ${\displaystyle T=M/\Gamma ,}$ 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 ${\displaystyle M}$ by any homothety of ratio different from ${\displaystyle \pm 1}$.

It can be verified that the curve

${\displaystyle \sigma (t):=\left({\frac {1}{1-t}},0\right)}$

is a geodesic of M that is not complete (since it is not defined at ${\displaystyle t=1}$).[1] Consequently, ${\displaystyle M}$ (hence also ${\displaystyle T}$) is geodesically incomplete, despite the fact that ${\displaystyle T}$ is compact. Similarly, the curve

${\displaystyle \sigma (t):=(\tan(t),1)}$

is a null geodesic that is incomplete. In fact, every null geodesic on ${\displaystyle M}$ or ${\displaystyle T}$ is incomplete.

The geodesic incompleteness of the Clifton–Pohl torus is better seen as a direct consequence of the fact that ${\displaystyle (M,g)}$ 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

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

consider

${\displaystyle F:N\to M}$

${\displaystyle F(u,v):=(\tan(u),\tan(v)).}$

The metric ${\displaystyle F^{*}g}$ (i.e. the metric ${\displaystyle g}$ expressed in the coordinates ${\displaystyle (u,v)}$) reads

${\displaystyle {\widehat {g\,}}={\frac {du\,dv}{{\tfrac {1}{2}}(\cos(u)^{2}\sin(v)^{2}+\sin(u)^{2}\cos(v)^{2})}}.}$

But this metric extends naturally from ${\displaystyle N}$ to ${\displaystyle \mathbb {R} ^{2}\smallsetminus \Lambda }$, where

${\displaystyle \Lambda =\left\{{\tfrac {\pi }{2}}(k,\ell )\ \mid \ (k,\ell )\in \mathbb {Z} ^{2},k+\ell \equiv 0{\pmod {2}}\right\}.}$

The surface ${\displaystyle (\mathbb {R} ^{2}\smallsetminus \Lambda ,{\widehat {g\,}})}$, known as the extended Clifton–Pohl plane, is geodesically complete.[3]

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 ${\displaystyle (-\pi /2,\pi /2)^{2}}$ i.e. "at infinity" in ${\displaystyle M}$. Recall also that, by Hopf–Rinow theorem no such tori exists in the Riemannian setting.[4]