Parallel transport
The operation that moves a vector along a curve while keeping it as parallel as possible — its covariant derivative along the curve vanishes: D V^μ/dλ = 0. On flat space this returns the same vector; on a curved space it generally rotates. The rotation around a closed loop is the holonomy and is a direct measure of curvature.
Definition
Parallel transport is the operation of carrying a vector along a curve on a manifold while keeping it as parallel as possible to itself in the local geometric sense — that is, demanding that its covariant derivative along the curve vanish: D V^μ/dλ ≡ (dx^ν/dλ) ∇_ν V^μ = (dV^μ/dλ) + Γ^μ_{νρ} (dx^ν/dλ) V^ρ = 0. The condition is a first-order ordinary differential equation for the components V^μ(λ) along the curve, and given an initial vector at one end, integration produces a uniquely transported vector at every point along the curve. The transport depends on the path between two points, not just on the endpoints — except on flat space, where parallel transport is path-independent and reduces to ordinary translation of vectors.
On a curved manifold, parallel transport along different paths between the same two endpoints generally gives different vectors at the endpoint. More dramatically, parallel-transporting a vector around a closed loop generally rotates it: the rotation angle is the holonomy, and its non-vanishing is a direct measure of curvature. The classic illustration is on a 2-sphere: a vector parallel-transported around a spherical triangle picks up an angular deficit equal to the enclosed area divided by R², which is the Gauss-Bonnet relation for that geometry. Parallel transport, defined intrinsically by Levi-Civita in 1917, is the geometric content of the Christoffel symbols and the conceptual root from which the Riemann curvature tensor is derived.