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Let $S$ be a smooth surface embedded in $\mathbb{R}^3$. When are (some of) the principal lines of curvature geodesics on $S$? Perhaps on the ellipsoid below, the (blue) central cycle, a max principal line, is a geodesic? And perhaps the (red) min principal line connecting the two umbilical points is a geodesic?

         
          Image from Jorge Sotomayor.1
Is there any $S$ all of whose principal lines of curvature are geodesics?

1Sotomayor, Jorge. "Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in ${\mathbb R}^ 3$." arXiv Abstract (2004). São Paulo Journal of Mathematical Sciences 2, 1 (2008), 99–143.

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    $\begingroup$ If a geodesic $\gamma$ is a line of curvature then it lies in a plane that perpendicular to the surface at any point of $\gamma$. This is quite rare thing --- most surfaces do not have such planes. $\endgroup$
    – Anton Petrunin
    Apr 15, 2019 at 17:29
  • $\begingroup$ @Joseph O'Rourke Intuitively by symmetry there can be only these three central cases. $ (x/a)^2+ (y/b)^2=1,\,(y/b)^2+ (z/c)^2=1,\,(z/c)^2+ (x/a)^2=1 $. Apart from it.. from Sotomayor's article or elsewhere do we have parametrization for the beautiful lines of curvature (on the Monge's ellipsoid)? $\endgroup$
    – Narasimham
    Apr 15, 2019 at 21:13
  • $\begingroup$ @AntonPetrunin: Thanks, Anton, that's a useful observation! $\endgroup$
    – Joseph O'Rourke
    Apr 15, 2019 at 21:14

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In your example: Yes, because these are fixed point sets of reflections which induce isometries on the ellipsoid.

All principal lines are geodesics on circular cylinders, again as fixed point sets of reflections.

Also other cylinders (like an oval in $\mathbb R^2$ times $\mathbb R$) have this property, because they are ruled, so the straight lines are geodesics, and the other principal lines are perpendicular to these, and they are fixed point of reflections, so they are also geodesics.

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