Affine connexion

Parallel transport

In an affine space, the “tangent space” at every point is identical, and encodes translations. Thus we may freely transport vectors based at a point to vectors based at a point , maintaining parallelism. For a “locally affine” space — a -manifold — this is made possible by the data of an affine connexion.

Definition

Let be a -curve with tangent vector , and let be an assignment of a vector at each point along the curve. We say that is parallelly transported along iff diff

all along the curve. Choosing local coördinates and taking the connexion coëfficients this becomes

or in components

It follows from the Existence and uniqueness theorem for IVPs that the parallel transport of a vector along a given curve is unique.

Remarks

• The Levi-Civita connexion is chosen precisely so that it is torsion-free and the inner product of vectors is preserved as they are parallelly transported together.

---

tidy | en | sembr