Moving vectors along curves and the geometry of loops
Parallel transport is the process of moving a vector along a curve while keeping it “as parallel as possible” — meaning it doesn't rotate relative to the surface.
On flat surfaces, this is trivial. On curved surfaces, something remarkable happens: a vector transported around a closed loop returns rotated! This rotation is called holonomy.
Watch the orange vector being parallel transported along a geodesic. The ghost vectors show its history — notice how it rotates even though it's maintained parallel at each infinitesimal step!
Watch the orange vector being parallel transported along the curve. The ghost vectors show its history — notice how it rotates even though it's being kept “as parallel as possible” at each step!
A vector V is parallel along a curve γ(t) if its covariant derivative vanishes:
In coordinates, this becomes a system of ODEs involving the Christoffel symbols, similar to the geodesic equation but for the vector components.
The classic demonstration: transport a vector around a spherical triangle. The holonomy angle equals the spherical excess(area on a unit sphere). Adjust the triangle vertices to see the relationship!
Gauss-Bonnet: When you parallel transport a vector around a closed loop, it rotates by an angle equal to the enclosed Gaussian curvature × area. On a unit sphere (K = 1), this equals the spherical excess!
The holonomy around a closed loop is intimately connected to the Gaussian curvature enclosed by that loop:
This is a local version of the Gauss-Bonnet theorem. The total rotation equals the integral of Gaussian curvature over the enclosed region!
Select different loops on various surfaces to compute their holonomy angles. Compare loops in regions of positive vs negative curvature!
The holonomy angle is the total rotation of a parallel-transported vector after traversing a closed loop. It equals ∫∫R K dA by the Gauss-Bonnet theorem.
Positive holonomy — vectors rotate in the same direction as the loop traversal.
Zero holonomy — vectors return unchanged after any closed loop.
Negative holonomy — vectors rotate opposite to the loop direction.
Parallel transport is governed by the connection, which tells us how to “connect” tangent spaces at nearby points. In an orthonormal frame, this is captured by the connection 1-form ω:
The curvature is the exterior derivative: K dA = dω. This connects the local (infinitesimal) rotation rate to the global (holonomy) rotation.
Next: Minimal Surfaces — surfaces with zero mean curvature, like soap films.