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This is a repost of a question I posted at MSE.

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 "ergodic": aperiodic and covering either the entire surface - by spiraling endlessly around - 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.

Does the structure of geodesics change when twisting the "hose" before gluing its ends?

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E.g., there might be no equators anymore because after twisting the (two) equators lost their "ends".

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  • $\begingroup$ I did not understand the question but want to say something. $$ $$ If the torus can be is fibered by closed geodesics then any other geodesic has to go transverally to the fibers. I think that is the only idea behind any statement in this direction; i.e., if it does not help then nothing will help. $\endgroup$
    – Anton Petrunin
    Commented Oct 23, 2012 at 22:39
  • $\begingroup$ BTW, the equator will move but it will survive. $\endgroup$
    – Anton Petrunin
    Commented Oct 23, 2012 at 22:40
  • $\begingroup$ @Anton: Where do the equators go? How do they survive? $\endgroup$
    – Hans-Peter Stricker
    Commented Oct 23, 2012 at 22:49
  • $\begingroup$ @Anton: To be honest, I have nothing overly specific in mind when asking for "the structure of geodesics". It's partly about the classification of geodesics - does it still hold after a twist? -, and it's partly about something comparable to the cycle space of a graph (en.wikipedia.org/wiki/Cycle_space). $\endgroup$
    – Hans-Peter Stricker
    Commented Oct 23, 2012 at 23:09
  • $\begingroup$ Take a marker, draw a geodesic curve on the torus, then do a Dehn twist. The resulting curve is a geodesic as the geodesic equation is local. With this you can easily see how the families above map to one-another. $\endgroup$
    – Kelly Davis
    Commented Oct 24, 2012 at 16:06

2 Answers 2

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It is the same submanifold of $\mathbb R^3$ with same geodesics. You are just describing another chart for it which is open and dense. Another way to say this and to visualize the geodesics in this other chart is:
Lift each geodesic to the universal cover $\mathbb R^2$ and use a parallelogram instead of a square as fundamental domain.

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  • $\begingroup$ This is not a flat torus, it has curvature. $\endgroup$
    – Will Sawin
    Commented Oct 27, 2012 at 19:31
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    $\begingroup$ Sure, the geodesics will not be straight lines in the universal cover. Nevertheless you can lift them. $\endgroup$
    – Peter Michor
    Commented Oct 29, 2012 at 7:55
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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.

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