 Distance splitting through a small geodesic sphere

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.