= Distance splitting through a small geodesic sphere
{title2=$d(p,q)=r+d(p_0,q),\quad d(p,p_0)=r$}
In a connected <Riemannian manifold>, choose a small compact <geodesic sphere> around $p$ with $0<r<d(p,q)$. Minimize $d(\cdot,q)$ on this sphere. Every path from $p$ to $q$ crosses it, so an almost-minimizing path has length at least $r+\min_{S_r(p)}d(\cdot,q)$. The <triangle inequality> gives the reverse bound, yielding the displayed equality. No global completeness is needed.
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