Geodesics on the torus [This is a follow-up to my question Is there a Möbius torus?]
Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: 


*

*There are five clear-cut families of geodesics.

*Most of the geodesics are "chaotic": aperiodic and covering either the entire surface - by spiraling endlessly around the torus - or substantial parts of it.

*Some of the geodesics are "boring": the meridians, the inner and the outer equator

*A few of them are "æsthetically pleasing": returning to their starting point after just a few circuits around the z axis 


What I tried to ask in my previous question: 

Can the structure of geodesics on the torus change drastically when
  twisting the "hose" before gluing its ends?

For example: There might be no equator anymore because after twisting the (two) equators lost their "ends".
[I also posted this question at MO.]
 A: My article directly compares the spectrum of closed geodesics on the flat torus to the normal torus:
http://www34.homepage.villanova.edu/robert.jantzen/notes/torus/
By thinking like a physicist, you can get a much better picture of this problem.
A: Lets say the Torus was painted with stripes parallell to the long axis before the twist. If the twist was just enough to put the beginning of the first line and the end of the second line together, you would get one continuous line all the way, I Think. Except if the twist was 180 degrees (or some multiple thereof), and there was an even number of lines, in which case you would get two (or pairs) of continous lines, each with half of the lines in a row. 
A: If the twisting angle is some rational number of full $2\pi$ twists, say $\frac pq$ revolutions, $p,q\in\bf \mathbb{Z}\space\backslash \left\{0\right\}$, then your equator ends will reconnect together after exactly $\frac qp$ full revolutions.
If it's irrational number of full-twists, then the resulting equator orbit will cover the whole torus space, and have Hausdorff dimension two, instead of ordinary one.
Based on irrational rotation dynamical systems.
