##  [Holonomy (of a Connection)](/holonomy-connection-0) 

 Definition

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

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

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

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

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

Reversal

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

 

 

 

 

 





## Boundary

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

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

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.