Holonomy
The rotation a vector picks up when parallel-transported around a closed loop on a manifold. Vanishes if and only if the curvature is zero. On a 2-sphere of radius R, transporting around a loop enclosing area A rotates the vector by A/R² — the spherical-triangle holonomy reveal that motivates the Riemann tensor.
Definition
Holonomy is the linear map produced on the tangent space at a point by parallel-transporting vectors around a closed loop based at that point. On a flat manifold the holonomy of every loop is the identity — vectors come back to themselves regardless of path. On a curved manifold the holonomy is non-trivial: parallel transport around a loop generally rotates a vector by a non-zero angle, and the rotation depends on the geometry enclosed by the loop. The set of all holonomies forms the holonomy group of the manifold, a subgroup of the orthogonal group O(n) for a Riemannian metric (and of the Lorentz group for a Lorentzian one). The holonomy vanishes for every loop if and only if the manifold is flat.
The classical illustration lives on the 2-sphere of radius R. Parallel-transporting a vector around a spherical triangle whose vertices are the north pole and two points on the equator produces a rotation equal to the area A enclosed divided by R². This is the spherical-triangle "money shot" reveal — the holonomy of an infinitesimal loop is exactly the Riemann curvature tensor contracted against the loop's bi-vector area element, which is how Riemann curvature is sometimes most concretely defined: R^ρ_{σμν} measures the holonomy per unit area in the (μ, ν) plane acting on a vector with σ and ρ indices. Curvature is what holonomy detects, and holonomy is what makes curvature locally measurable without leaving the manifold.