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'?