Let \((M,g)\) be a Riemannian manifold. Then there is an open neighborhood \(U\subset M\times M\) of the diagonal \(\{(x,x)\;\;:\;\;x\in M\}\) and a smooth map \(\gamma:U\times[0,1]\to M\) such that for all \(p,q\in U\) the curve \(\gamma_{p,q}:[0,1]\to M\) is the unique minimizing geodesic between \(p,q\).