##  [Parallel Transport](/parallel-transport-0) 

 Definition

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

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

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

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

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

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

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

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

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.