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

2025/10/11 by Pinteaux, Constant, Tuynman, Gijs M.
#53A99 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.10247

Abstract

Given a submanifold M⊂ R^ν, a curve γ:I→ 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.

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