I'm reading a snapshot of what modern mathematics looks like: Closed geodesics on surfaces and Riemannian manifolds.
Define$$ O^2:=E^1 \times E^1 $$ where $$ E^1:= \big\{(u,v)\in\Bbb R^{2+}:\log^2(u)+\log^2(v)=1\big\}. $$
Here $\Bbb R^{2+}$ is equipped with the metric $ds^2=\frac{du^2}{u^2}+\frac{dv^2}{v^2}.$
How can one find the closed geodesics on $O^2?$ (not including meridian and longitude geodesics).
Credits to @robjohn for these:
I know that there must be at least one closed geodesic, and I think there could be infinitely many closed geodesics due to a theorem:
Theorem 1: (Gromoll–Meyer, Sullivan–Vigué-Poirrier) Any Riemannian manifold whose rational cohomology ring is generated by at least two elements contains infinitely many distinct closed geodesics.
Also the meridian and longitude geodesics on $O^2$ are (under a coordinate change) equivalent to the meridian and longitude geodesics on $T^2$ (torus). Could this be the case for all the geodesics? That is, could all the geodesics on $O^2$ be related by a coordinate change to the geodesics on $T^2?$
I don't understand this question. This space is just the torus $T^2,$ and so has the usual complement of geodesics.