The group of parallel-transport transformations obtained by transporting vectors (or frames) along closed loops in a manifold with a given connection; encodes curvature and global geometric constraints.

Principle

Parallel transport around loops composes connection-dependent linear maps; the set of all such maps based at a point forms the holonomy group, whose structure reflects curvature and topology.

Demonstration

On a sphere with the Levi-Civita connection, parallel transporting a tangent vector around a geodesic triangle yields a rotation whose angle equals the triangle's spherical excess — a manifestation of nontrivial holonomy.

Misapplication

Treating holonomy as a local tensor field value; holonomy is a global group of transformations and cannot be specified by pointwise values alone without path data.

Consequence

Nontrivial holonomy constrains the existence of parallel sections, special metrics, or reduced structure groups (e.g., special holonomy manifolds admit covariant constant spinors), affecting global geometry and physics.

Reversal

Trivial holonomy (holonomy group reduced to identity) implies flatness of the connection on simply connected domains and existence of globally parallel frames.

Boundary

Defined for connections on principal or vector bundles over manifolds; does not apply to arbitrary transport rules lacking compatibility with a linear connection structure.

Semantic Tension

Sometimes confused with curvature tensor components; curvature is the infinitesimal generator of holonomy, while holonomy collects finite parallel-transport effects around loops.

Synthesis

Holonomy is the group of linear transformations produced by parallel transport along closed loops for a given connection, capturing global curvature and geometric constraints.