Prove that:

Constant geodesic curvature lines on constant Gauss curvature surfaces are closed Ovals/Loops.

Find perimeter/length of this Oval/Loop in terms of $ k_g$ and $K$

I believe the proof runs directly from geodesic polar coordinates considered conversely, but for its exact manner I ask for help.


Also please find relationship between $ k_g$ and $ \sqrt { |K|}$ , identifying constant of integration to find its geometrical meaning/ context.

In particular case of a positive Gauss curvature sphere, $ k_g$ = $tan \gamma \sqrt { |K|}$ where $\gamma$ is brought in as the angle between arc normal and surface normal (i.e., latitude of parallel circles) that vanishes for geodesics.

The question is asked during my attempt to associate $ k_g$with normal curvature as a $k$ component in general:

$$ k_g = tan \gamma * k_n , k^2 = k_g^2 + k_n^2 $$

My logic is simple. I have visualized it in the following way. By Minding's Theorem all surfaces of equal constant $K$ are applicable over one another.So by continuous isometric mappings we can make all Ovals of this set pass through a given point on one (or for that matter any) of the surfaces conserving the same perimeter length.The point can then serve as a fulcrum as well,we may then be allowed to say in mathematical parlance "upto isometric bendings or mappings" in this $ (k_g, K) $context.

In other words, the oval is freely rotatable about a fixed point.Each Oval bends and twists without stretching during any surface contact application dragging its surface neighborhood with it while turning about this fulcrum in the tangent plane or moving it about, somewhat like a flexible magnetic oval patch moving on steel pseudosphere.

Below are given rotated Loops on pseudosphere.The loops/ovals are closed and appear to be continuously differentiable.

kg and K common Proceeding from these considerations what is sought in isometry is to obtain:

  • the metric or first fundamental form containing $ k_g$ and $K$ either implicitly or explicitly, and,
  • the derived differential equation containing $ k_g$ and $K$ explicitly as constants.


It is possible to see the constant $k_g$ lines including geodesic $ k_g=0 $ lines on wooden models drawn on spheres and pseudo-spheres in the text-book of Lectures on Classical Differential Geometry by D.J. Struik ( Dover 2nd Edition), Figures 4-11 and 4-12 on pp 149/150. He displays the $ K > 0, K < 0 $ 3D solid surface visualization here but does not mention so much about curves shown /drawn on them .. that in fact are more interesting and require more mathematical computation and careful transfer onto the 3D model while doing it.. iirc it is from a Mathematical museum in Munich, but no further historical detail is available.

We can see concentric geodesic polar curves for sure.

Hope someone researches about making of these lines on the models.


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