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 $𝑝 1$ to vectors based at a point $𝑝 2$ , maintaining parallelism. For a “locally affine” space — a $𝐶 𝛼$ -manifold — this is made possible by the data of an affine connexion.

Definition

Let $𝛾 : 𝐼 \to 𝑀$ be a $𝐶 𝛼$ -curve with tangent vector $˙ 𝛾 𝑎$ , and let $𝑣 𝑎$ be an assignment of a vector at each point along the curve. We say that $𝑣 𝑎$ is parallelly transported along $𝛾$ iff #m/def/geo/diff

˙ 𝛾 𝑎 \nabla 𝑎 𝑣 𝑏 = 0

all along the curve. Choosing local coördinates $𝑥 : 𝑈 \to ℝ 𝑚$ and taking the connexion coëfficients this becomes

˙ 𝛾 𝑎 𝜕 𝑎 𝑣 𝑏 + ˙ 𝛾 𝑎 Γ 𝑏 𝑎 𝑐 𝑣 𝑐 = 0

or in components

˙ 𝑣 + ˙ 𝛾 𝜇 Γ 𝜈 𝜇 𝜆 𝑣 𝜆 = 0 .

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.

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