# 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.]

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Topologically, the result of a simple twist will again be a torus. And a torus has an equator, and all the different classes of geodesics. However, the geodesics of the new torus are usually unrelated to those of the original torus: the twist introduced a bending to the whole metric, and would cause the old metric to become disconontinuous along the seam. So I cannot think of a way to reasonably answer this question. – MvG Oct 24 '12 at 9:15
Is there really a seam? And is there a seam in the Möbius strip? And is there a discontinuity? – Hans Stricker Oct 24 '12 at 12:12
If you claim that equators lost their “ends”, then in that interpretation there is a seam, where things that used to be geodesics in the untwisted gluing are geodesics no more. Usually I'd say there is no seam in the Möbius strip, but with that same argument I'd say a twisted torus is just the same as an untwisted torus. The argument here is that you concentrate on the topology of the thing up front, and obtain a compatible metric from that topology alone, without regard to any metric the thing might have had before twisting and/or gluing operations. – MvG Oct 24 '12 at 12:33
The Villarceau Circle, not a boring case but not a geodesic. – Narasimham Mar 10 '15 at 9:00