Moving vectors along curves and the geometry of loops
Carry an arrow around a loop on a curved surface, never letting it turn. It comes home pointing somewhere new. That twist, called holonomy, is curvature you can see.
Watch the orange vector ride along a geodesic, with ghost copies marking where it has been. Notice it rotates even though it is kept parallel at every 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!
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.
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.
Move the triangle's vertices and transport a vector around it. The holonomy angle always matches the triangle's spherical excess, its area on a unit sphere.
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!
Pick a surface and a loop to compute its holonomy angle. Compare loops where the curvature is positive, negative, and zero.
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.
Spot a mistake or have an idea?