Clifton-Pohl Torus and $\Gamma$ acting properly discontinuous

Problem: Let $$M = \mathbb{R}^2 \setminus \left\{ (0,0 \right\}$$ be the pseudo-Riemannian manifold with metric $$ds^2 = \frac{2 du dv}{ u^2 + v^2}.$$ Let $$\mu(u, v) = (2u, 2v)$$. This is an isometry (trivially). I wish to show that the group $$\Gamma = \left\{ \mu^n \right\}$$ generated by $$\mu$$ acts properly discontinuous on $$M$$. Then $$T = M/\Gamma$$ is a Lorentz surface, called the Clifton-Pohl Torus.

So by definition I need to prove that: every $$p \in M$$ has a neighborhood $$U$$ such that $$\phi(U) \cap U = \emptyset$$ for all $$\phi \in \Gamma$$ with $$\phi \neq Id$$. (do Carmo, Riemannian Geometry, p. 165).

The way I wished to show this is as follows. If $$p = (u_0, v_0) \in M$$, then let $$\epsilon = d((u_0, v_0), (2u_0, 2v_0))$$ where $$d$$ denotes the distance between $$(u_0, v_0)$$ and $$(2u_0, 2 v_0)$$ for the metric $$ds^2$$ on $$M$$. Then if I take as a neighborhood $$U = B_{\epsilon/2} (p)$$ , i.e. the 'ball' centered at $$p$$ with radial distance $$\epsilon/2$$, then any action of a $$\phi \in \Gamma$$ will move all of the points of $$U$$ out of $$U$$, i.e. $$\phi(U) \cap U = \emptyset$$.

The problem that I have is that I'm not sure if $$\epsilon > 0$$, since in a semi-Riemannian manifold the distance between two distinct points can be zero.

Moreover, does one define distance in Lorentzian geometry the same way as in Riemannian geometry? In Riemannian geometry the Riemannian distance $$d(p,q)$$ between two points $$p$$ and $$q$$ is defined as the greatest lower bound of $$\left\{ L(\alpha): \alpha \in \Omega(p, q) \right\}$$, where $$\Omega(p,q)$$ is the set of all piecewise smooth curve segments in $$M$$ from $$p$$ to $$q$$ and $$L(\alpha)$$ denotes the arc length of $$\alpha$$.

Does this definition extend to Lorentzian geometry?

• What you wrote on the 2nd line is a Riemannian metric, not Lorentzian one. Did you mean to use $u^2-v^2$ or $uv$ in the denominator? In any case, do not use the Lorentzian metric to define your fundamental domain, use the Euclidean annulus $\{(u,v): 1\le u^2+v^2\le 4\}$. The thing is that Lorentzian metrics are not good for metrizing topological spaces. – Moishe Kohan Apr 13 '19 at 13:07
• No I didn't mean $u^2 - v^2$; I meant exactly what I wrote. See page 193 of the book of O'Neill : Semi-Riemannian geometry. – Kamil Apr 15 '19 at 7:31
• Oh, you are right of course. The rest of my comment stands as is. – Moishe Kohan Apr 15 '19 at 23:25

The distance used in the definition of a properly discontinuous group action should be compatible with the topology on of the manifold. The Lorentzian distance is the wrong thing to use here. Since the group acts on $$\mathbb{R}\setminus\{(0,0)\}$$ you should use the euclidean distance, $$\rho$$.
Then given any point $$(u,v)$$ we can take $$U=\{(x,y): \rho((u,v),(x,y))<1\}$$. In your answer you seem to know what to do at this point to check the definition.