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

TL;DR

The paper reinterprets geodesics and parallel transport through a rolling-tangent-space construction: rolling the tangent space along a curve on $M$ produces a trace curve in a fixed affine tangent space, and the original curve is a geodesic precisely when this trace is a straight line. Parallel transport is realized as ordinary parallel transport in the rolling space, making the covariant derivative coincide with the trace-derivative, up to the isometry $\operatorname{P}_{\mathrm{T}}(t,s)$. The authors develop existence/uniqueness results for the rolling space evolution, prove the central equivalence between straight-trace curves and geodesics, and derive coordinate formulations that yield the standard geodesic equations in a local chart. This approach offers both a vivid geometric intuition and a concrete computational bridge between intrinsic Riemannian geometry and an affine rolling viewpoint, with explicit expressions in local coordinates via Christoffel symbols.

Abstract

Given a submanifold $M\subset \mathbf{R}^ν$, a curve $γ:I\to M$ and tangent vectors $v$ along $γ$, we roll the tangent space along $γ$. In doing so, we get an imprint of $γ$ on the tangent space, as well as an imprint of tangent vectors. We show that $γ$ is a geodesic on $M$ if and only if this trace/imprint on the (affine) tangent space is a straight line and that $v$ is a set of parallel vectors if and only if their imprint on the tangent space is constant. In other words, in the view of the imprint on the rolling tangent space, a geodesic is a straight line, parallel transport is indeed that: parallel transport, and the covariant derivative becomes the ordinary derivative.