Show that the set of all straight lines in $\mathbb{R}^{2}$ admits the structure of an abstract smooth surface $S$. Show that $S$ is an open Möbius band (i.e. the Möbius band without its boundary circle), and deduce that $S$ admits a Riemannian metric with vanishing Gauss curvature.

Show that there is no metric $d: S \times S \rightarrow \mathbb{R}_{\geqslant 0}$, in the sense of metric spaces, which

1. induces the locally Euclidean topology on $S$ constructed above;

2. is invariant under the natural action on $S$ of the group of translations of $\mathbb{R}^{2}$.

Show that the set of great circles on the two-dimensional sphere admits the structure of a smooth surface $S^{\prime}$. Is $S^{\prime}$ homeomorphic to $S$ ? Does $S^{\prime}$ admit a Riemannian metric with vanishing Gauss curvature? Briefly justify your answers.