Paralleltransport

In differential geometry, parallel transport is a way of moving a tangent vector along a curve on a manifold so that it remains as "parallel" as possible with respect to a given affine connection. It provides a means to compare vectors at different points in a coordinate-free way.

Let M be a smooth manifold with an affine connection ∇. A vector field V(t) along a curve γ(t) is parallel if ∇_{\dot γ} V = 0. Given an initial vector V(t0) = v0, there is a unique parallel transport of v0 along γ from t0 to t1, yielding V(t1). In local coordinates, with γ(t) = (x^i(t)) and V(t) = V^i(t) ∂/∂x^i, the equation reads dV^i/dt + Γ^i_{jk}(x(t)) \dot x^j V^k = 0. The solution can be written formally as a path-ordered exponential of the connection along γ.

If the connection is metric-compatible, as in a Riemannian manifold with the Levi-Civita connection, parallel transport preserves inner products: ⟨V(t), W(t)⟩ is constant along γ. Consequently, lengths and angles of transported vectors are preserved.

The nontrivial effect of parallel transport around loops is governed by curvature; transporting a vector around a closed loop generally returns a differently oriented vector. The collection of all such transformations forms the holonomy group of the connection.

Examples and applications: in Euclidean space with the standard flat connection, parallel transport is trivial and vectors stay constant along any curve. On a sphere with the Levi-Civita connection, transporting a vector around a loop yields a rotation related to the loop’s area. Parallel transport plays a central role in gauge theories and general relativity, where it describes how fields are compared across spacetime; geodesics are curves whose tangent vectors are parallel transported along the curve itself, satisfying ∇_{\dot γ} \dot γ = 0.

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