I have been having the darnedest fun trying to understand geodesics on the surface z=x^2 + y^2. It turns out I got all the way through graduate school, including courses in differential geometry, without realizing that the book examples were cooked up to be do-able and that, for the most part, even simple expressions lead to unanswerable questions. For example you study curves using "arc-length" parametrization when, in simple cases like (t, t^2, t^3), the arc-length parametrization cannot be found easily.

Anyway, the geodesics on z=x^2 + y^2 that do not go through (0,0,0) can only spiral outward and will always have an infinite number of windings around the z axis. I don't know why it took a couple months to realize a geodesic cannot spiral inward without its curvature exceeding that of the surface and its velocity vector is always turning more in the horizontal direction than in the vertical direction - so it integrates faster in theta than in z - guaranteeing a winding. The lemma is that geodesic's velocity vector will always turn toward the direction of greatest bending of the surface.Also, I find it interesting that winding number is a topological property of the geodesics but not of the surfaces. You can deform a surface just slightly and change the winding numbers dramatically. The lemma suggest other things to look at, like z=a*x^2+b*y^2 where the greatest amount of bending can be a horizontal direction in some places and vertical direction in others.

Not sure that last bit about horizontal/vertical is actually possible.

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