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I have troubles trying to imagine the geodesic curvature of curves on surfaces of positive gaussian curvature. Not being generally valid the "minimum distance" argument, I have difficulty to grasp the intuition of this quantity even in very simple cases. I am not interested in explicit calculations, I would like to know what could be some sensible ways I should look at two different curves on a "beautiful" surface with positive gaussian curvature and try to predict which is "further" from being geodesic (even if I don't know how all the geodesic of the surface look like - i.e. I don't have a comprehensive classification that allows me to make sensible comparisions)?

More specifically, I want to know how to tell if the geodesic curvature of a parallel of a two-sheeted hyperboloid decreases as the parallel gets larger (i.e. as it gets further from the point of maximum gaussian curvature) without explicit calculation. Thinking of its definition as the norm of the covariant derivative doesn't seem to help much (maybe I'm just not confident enough with the covariant derivative?)...and using the fact that parallels through critical points of the curve whose rotation generates the surface are the only geodesic parallels doesn't seem to suggest the right answer (I think the geodesic curvature goes down to zero but the derivative of the generating curve does not...)? What goes wrong here? How should I go about it then?

I hope my question is sensible enough, though I realize it is fairly vague and imprecise! And of course, I am just looking for ideas to make sensible expectations that could work in "beautiful" cases with lots of symmetries like the one presented, nothing too general. It's just that it doesn't feel like I've fully understood it until I can make sense of the simple cases (like this one) without uggly calculations. Thanks in advance so much for taking your time to answer me!

P.S.: just so you know: I am un undergraduate student who has taken just one course of differential geometry.

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First, let $c\colon I\subset\mathbb{R}\longrightarrow S$, be a geodesic of a regular surface $S$, parametrized by arc length. If $N$ denotes the unit normal of $S$, then recall that $N$ is parallel along $c$ to the principal normal $\vec{n}$ of $c$, i.e. $$N(c(s))\parallel \vec{n}(s),\ \forall s\in I.$$ Therefore, if $c_t$ denotes the parallel of the two-sheeted hyperboloid of height $\vert t\vert$, (i.e. $c_t$ is the circle which is the intersection of the plane $z=t$ with the two-sheeted hyperboloid [of course only for the $t$ that the intersection is defined]) then as $t$ goes to infinity the normal approaches the principal normal of the curve $c_t$. Thus $c_t$ tends to be a geodesic and thats why the geodesic curvature of the parallel $c_t$ decreases as $t\rightarrow \infty$.

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Since you say no explicit calculations, I shall be brief towards conceptualization. You need to trace lines of constant geodesic curvature.

The topic is "geodesic polar coordinates & geodesic parallels". $ k_g = 1/u ;\,$ $ u = const. $ are "radial " lines. $v= const. $ are concurrent geodesics at "Bull's Eye,$u=0$". The concentric rings are geodesic parallels.

All parallels of a 2 sheeted axisymmetric hyperboloid are having " constant minimum distance " between neighboring rings.

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