This is an extensive re-write of a question I deleted and which had basically the same title.
Identify the cylinder $S^1 \times \mathbb{R}$ with the space of (co)oriented lines in the plane by associating to a point $(\mathbf{u},p) \in S^1 \times \mathbb{R}$ the (co)oriented line $\mathbf{u} \cdot \mathbf{x} = p$.
By the work of Busemann, Pogorelov, Alexander, and Ambartzumian, it is known that continuous distance functions on the plane for which geodesics are straight lines are in a simple geometric correspondence with Borel measures on $S^1 \times \mathbb{R}$ satisfying the following properties.
- The measure is bundleless: the measure of all lines passing through a point in zero. This is the same as asking that the sets $$ \{(\mathbf{u}, \langle \mathbf{u},\mathbf{x} \rangle) : \mathbf{u} \in S^1 \} \subset S^1 \times \mathbb{R} $$ have measure zero for all $\mathbf{x} \in \mathbb{R}^2 $.
- The measure of any open set is positive.
- The measure of any compact set is finite.
If we have such a measure, then we define the distance between two points as the measure of all lines separating them and that gives us a metric whose geodesics are straight lines.
If we want to find such metrics on the two-torus, we also ask that the measure be invariant under the induced action of $\mathbb{Z}^2$ on the space of oriented lines, which is simply $$ ((m,n), (\cos(\theta),\sin(\theta); p)) \longmapsto ((\cos(\theta),\sin(\theta); p + m\cos(\theta) + n\sin(\theta)) . $$
If the measure is absolutely continuous with respect to the Lebesgue measure on the cylinder this implies that it is invariant under all translations and so the metric obtained from it will be nothing but one coming from a norm.
Question. What are (or what can be said about) all the singular measures on the cylinder that satisfy (1) and (3) while being invariant under the $\mathbb{Z}^2$ action given above?
There is a simple way to construct some of them by adding a finite or countable number of measures of the form $$ \langle \mu | \varphi \rangle := \int_{-\infty}^\infty \varphi(\frac{(m,n)}{\sqrt{m^2 + n^2}};p)f(p)\, dp $$ where $(m,n)$ is an integer vector and $f$ is a locally integrable function which is periodic of period $\gcd(m,n)/\sqrt{m^2 + n^2}$.
Is that it or is there something more exotic?
In the deleted question I seriously botched up the translation to analysis (to the cosine transform). I'm really sorry about that.
