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Can geodesics on any surface ( smooth, embedded $\mathbb R ^3)$ be mapped as straight lines on the plane? Please provide examples.

Are there some other global properties of surfaces that may need to be satisfied for a plane straight line representation ?

Stereographic projection of great circles of a sphere clearly do not map to straight lines on the plane. Longitudes of a sphere are obvious maps but may be trivial examples.

Not clear if any isometries need be modified.

Appreciate all helpful suggestions.

Narasimham
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  • Surely you can always map at least a substantial portion of at least one geodesic as a straight line. And just as surely (example already given in the question) there are surfaces for which there is no mapping in which all geodesics map to straight lines. Is there some in-between property you're looking for? – David K Feb 26 '24 at 05:07
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    Thanks for comment. Edited the question. I was looking to any global properties needing to be satisfied. – Narasimham Feb 26 '24 at 05:18
  • @ David K: Somewhere I remember to have read that Eugino Beltrami during the early times when hyperbolic geometry was gaining ground.. that constant Gauss curvature surfaces satisfy the condition of straight line nettings of Sine Gordon ( which are definitely geodesics in their own right ) and the plane answer the question I have here. I am of the same opinion, but if I state it there would be a salvo down-voted by those who may have authored articles or books in differential geometry. – Narasimham Feb 26 '24 at 15:45
  • contd.. .Not that reputation is so important but my aim as well as the purpose of this website is finding out relevant facts. Thanking you for pondering over these quite important matters. – Narasimham Feb 26 '24 at 15:46
  • OK, I understand the question now, I think. I don't have an answer as I'm a little out of my depth here. But I'm following the question now because you have gotten my interest. – David K Feb 27 '24 at 04:05
  • Thanks. I do not know if the reference found here is relevant: https://www.jstor.org/stable/1986354?seq=1 For constant K<0 which of two possibilities, hyperbolic asymptotes or usual geodesics ? considered with plane straight lines can be so mappable.. is to me not clear. – Narasimham Feb 27 '24 at 17:09

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