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Let $a,b,c\in\mathbb{R}^n$ such that $c$ is inside the $n$-disk with $a$ and $b$ as south and north poles. Then as $c$ moves toward $a$ through the line segment joining $a$ and $c$, $c$ is also moving away from $b$ (in terms of $d(b,c)$).

Is there an analogy of the above statement on Riemannian manifolds? Say replace 'distance' by 'Riemannian metric' and replace 'line segment' by 'geodesics'?

Or, in other words, fix any $a$ and $b$ on a Riemannian manifold, what is the set of points $c$ such that when $c$ moves to $a$ through geodesics, $c$ is moving away from $b$? On $\mathbb{R}^n$ we know the set contains all points $c$ such that $\angle acb$ is not acute.

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    $\begingroup$ Let your space be a line, $c=0,a=1,b=2$. When $c$ moves to $a$ through the line segment $[0,1]$, it is moving towards $b$. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 13, 2020 at 11:24
  • $\begingroup$ @AlexandreEremenko In this case $c$ is not inside the disk with $a$ and $b$ as south and north poles, where the disk is $[1,2]$. $\endgroup$
    – ryanriess
    Commented Jul 13, 2020 at 16:38
  • $\begingroup$ Then please explain what are the South and North poles for arbitrary Riemannian metric. $\endgroup$
    – Alexandre Eremenko
    Commented Jul 13, 2020 at 17:36
  • $\begingroup$ @AlexandreEremenko I am not sure, and that is what I am wondering. Put it in another way, fix any $a$ and $b$ on a Riemannian manifold, what is the set of points $c$ such that when $c$ moves to $a$ through geodesics, $c$ is moving away from $b$? $\endgroup$
    – ryanriess
    Commented Jul 13, 2020 at 17:51
  • $\begingroup$ You are talking about the Dirichlet domain / Voronoi diagram of a pair of points. It perhaps has other names, but that's one. $\endgroup$
    – Ryan Budney
    Commented Jul 13, 2020 at 18:54

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