A rule for moving tangent vectors along a curve in a differentiable manifold so that their covariant derivative along the curve vanishes relative to a chosen affine connection; it provides a way to compare vectors at different points by following the connection's prescription.

Principle

Preserve the connection-defined directional derivative: a vector field along a curve is parallel transported if its covariant derivative along the curve equals zero, ensuring no intrinsic 'twisting' relative to the connection as one moves.

Demonstration

On the two-sphere with the Levi-Civita connection of the round metric, transport a tangent vector along a closed triangle composed of great-circle arcs; the final vector typically differs from the initial one, showing holonomy related to the enclosed area.

Misapplication

Assuming parallel transport is path-independent in curved manifolds; except for flat connections, transporting along two different curves between the same endpoints can yield different resulting vectors.

Consequence

Defines holonomy and curvature phenomena: nontrivial parallel transport around loops encodes curvature information and leads to geometric phase effects in physics and differential geometry.

Reversal

Reversing the direction of the curve inverts the transport map along that curve; composing transport along a curve with transport along its reverse gives the identity when using the same connection.

Boundary

Depends on a chosen connection and applies to tangent (or associated) vector bundles over differentiable manifolds; it excludes arbitrary prescriptions that do not derive from a connection and cases where no smooth structure or connection is defined.

Semantic Tension

Closely related to 'covariant derivative' (local differential operator) and 'parallelism' (global notion); parallel transport is the integrated, pathwise realization of the covariant derivative but differs from naive coordinate-wise constancy or Euclidean translation.

Synthesis

Parallel transport is the connection-governed procedure that carries vectors along curves with zero covariant derivative, making the manifold's curvature manifest through path-dependent changes and holonomy.