Let \((M,g)\) be a Riemannian manifold and \(p\in M\). Let \(r>0\) such that \(\exp_p|_{B_r(0)}\) be a diffeomorphism on its image. Then the minimizing curve between \(p\) and \(q\in \exp_p (B_r(0))\) is given by the connecting geodesic [@lee2018riemannian_manifolds, Theorem 6.15].