For all \(p\in M\) and \(v\in T_pM\) there is a unique maximal geodesic \(\gamma\colon I\to M\), with \(0\in I\), \(\gamma(0)=p\) and \(\gamma'(0)=v\) [@lee2018riemannian_manifolds, Corollary 4.28].
Remark
- We may identify every \(v\in TM\) with a geodesic \(\gamma_v\). The point is determined by choosing the tangent space of \(v\).