Use of G\"odel Universe to Construct A New Zollfrei Metric with $R^2\times S^1$ Topology

A new example of $(2+1)$-dimensional Zollfrei metric, with the topology $R^2 \times S^1 $,ispresented\mathrm{.}Thismetricisreadilyobtainedfromthecelebrated$(3+1)$-dimensionalrotatingG\ddot{o}deluniverse$G_{3,1}$\mathrm{.}Thisisbecause$G_{3,1}$ has the interesting property that, the light rays which are confined to move on the plane perpendicular to the rotation axis, return to their origin after a time period $T=\frac{2\pi }{\omega }[\sqrt{2}-1]$ -where $\omega$ is the angular velocity of the universe. Hence by - the topological identification of pairs of points on the time coordinate, seperated by the time interval $T$. and droping the flat $x_3$ coordinate - which is directed along the rotation axis; one obtains the $(2+1)$-dimensional Zollfrei metric with the $R^2 \times S^1$ topology.