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What if we define a circle not as the set of points with the distance $R$ from the center, but rather as a set of points which have at least one geodesic path to the center of the length $R$?

Will the circles become much more complicated? Will the circles on toruses become everywhere dense after some radius?

Are there online visualizations of cirles defined this way on toruses and cones?

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  • $\begingroup$ On a sphere, since geodesics are great circles, the circles you are saying are still standard circles. In any case, for given center $x_0$ and radius $r$, it’s a compact (the image of the circle of $T_{x_0}M$ via the exponential map) $\endgroup$ Commented Dec 18, 2024 at 14:28
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    $\begingroup$ If by torus you mean the flat torus, then the radius $r$ circles you mention would just be the projection of a normal radius $r$ circle in $\mathbb{R}^2$ to the torus $\mathbb{R}^2/\mathbb{T}^2$. So indeed they tend to be everywhere dense, in the sense that for all $\varepsilon>0$ there is some radius $R$ such that any circle of radius $\geq R$ is at distance $<\varepsilon$ of all points in the torus. Something like $R=10/\varepsilon^2$ should work $\endgroup$ Commented Dec 18, 2024 at 15:00
  • $\begingroup$ @SaúlRM yes I meant everywhere dense as a limit with infinite radius... $\endgroup$ Commented Dec 18, 2024 at 15:03
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    $\begingroup$ In fact it looks like these circles should be everywhere dense in almost any compact manifold (the spheres would be an exception), if I get a nice result I will post it as an answer $\endgroup$ Commented Dec 18, 2024 at 15:05
  • $\begingroup$ @SaúlRM awaiting! Also, my motivation was to study similar spaces but with the opposite property - the greater radius, the less is the density. I believe, such spaces are negative-dimensional. $\endgroup$ Commented Dec 18, 2024 at 15:11

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