The following exercise is taken from M. do Carmo's Riemannian Geometry book
The problem is that I can't interpret it. In order that $P$ becomes an isometry, it should be a differentiable map between Riemannian manifolds. Despite $T_{c(t_0)} M$ and $T_{c(t)} M$ being isomorphic to $\mathbb{R}^{\dim M}$ (which is Riemannian manifold with Euclidean metric), there is no canonical identification I can find in the literature.