The rolling tangent space, a forgotten vision on parallel transport and geodesics

Given a submanifold $M\subset {\mathbf{R}}^{\nu }$, a curve $\gamma :I\to M$ and tangent vectors $v$ along $\gamma$, we roll the tangent space along $\gamma$. In doing so, we get an imprint/trace of $\gamma$ on the tangent space, as well as an imprint/trace of the tangent vectors. We show that for a vector field $v$ along $\gamma$, the imprint/trace of its covariant derivative is the ordinary derivative of its imprint/trace vector field. It then follows easily that $v$ is a set of parallel vectors along $\gamma$ if and only if their imprint/trace on the (affine) tangent space is constant and that $\gamma$ is a geodesic if and only if its trace on the tangent space is a straight line.