What rhumb lines of a torus are periodic $C^1$ curves? This is a question coming from an old (French) geometry book.
Take a 3D torus. Study the rhumb lines of the torus and find the ones that are periodic $C^1$ curves. In particular, it is mentioned that the meridian circles are not the only periodic rhumb lines.
I tried to write the equations of the rhumb lines, but I'm not able to find the periodic ones apart from the meridian circles. Do you have a clue on what could be those additional rhumb Lines?
For the record:


*

*The rhumb lines are the one crossing the meridian circles with a constant angle.

*The meridian circles are the one lying in planes containing the axis of revolution of the torus.

 A: I will assume the 3-D torus is obtained by rotating a circle of radius $b$ along a line at a distance $a > b$ from its center.
With a suitable choice of origin and coordinate axis, we can parametrize the 3-D torus as
$$[-\pi,\pi] \times [-\pi,\pi] \ni (\theta,\phi)
\quad\mapsto\quad ((a+b\cos\theta)\cos\phi, (a+b\cos\theta)\sin\phi, b\sin\theta) \in \mathbb{R}^3$$
In this parametrization, the metric is given by
$$b^2 d\theta^2 + (a+b\cos\theta)^2d\phi^2$$
It is clear the lines for fixed $\theta$ and $\phi$ are orthogonal to each.
For the rhumb line to make a constant angle $\alpha \ne 0$ with the meridian circles (i.e the lines for fixed $\phi$), one can read off from above metric following condition for the rhumb line.
$$\frac{b d\theta}{(a + b\cos\phi) d\phi} = \cot\alpha\tag{*1}$$
When one walks along a rhumb line and complete one cycle in the $\theta$ direction. The change in $\phi$ will be equal to
$$\Delta\phi_{1\text{cycle}} = \tan\alpha\int_{-\pi}^\pi \frac{b d\theta}{a + b\cos\theta}
= 2\pi\left(\frac{b \tan\alpha}{\sqrt{a^2-b^2}}\right)
$$
If $\displaystyle\;\frac{b\tan\alpha}{\sqrt{a^2-b^2}} = \frac{m}{n}\;$ is rational, then after one walks along a rhumb line and complete $n$ cycle in the $\theta$ direction, the total change in $\phi$ will be $2\pi m$ and the rhumb line meet itself. 
In short, the other rhumb lines are given by those angles $\alpha$ where
$\displaystyle\;\frac{b\tan\alpha}{\sqrt{a^2-b^2}}\;$ is rational.
If you want the actual equations for the rhumb lines, you can treat the condition $(*1)$ as an ODE and integrate it with change of variable $x = \tan\frac{\theta}{2}$. The resulting curve is not only $C^1$ but smooth (when $a > b > 0$). Furthermore, when $\displaystyle\;\frac{b\tan\alpha}{\sqrt{a^2-b^2}}\;$ is rational, the curve is periodic and smooth.
The actual computation is pretty simple and I'll leave that as an exercise.
A: You can measure the "longitude" of each point on the torus so that
for each $2\pi$ increase in longitude you return to the same meridian circle.
You can measure the "latitude" of each point on the torus so each
$2\pi$ increase in latitude along a meridian circle returns you to the same point,
and so that the points at a fixed latitude form a circle in a plane
perpendicular to the axis of rotation.
At any point along a rhumb line, you can compute the rate of change of
longitude per unit of change of latitude.
This quantity is a function of the latitude.
Now integrate the change in longitude over one full cycle of latitude
(such as from $-\pi$ to $\pi$).
If the change in longitude for a $2\pi$ change in latitude is a
rational multiple of $2\pi$, after a finite number of $2\pi$-long cycles
of latitude you will return to the same longitude.
