Holonomy

Holonomy is a concept in differential geometry that describes how parallel transport around a closed loop affects objects such as vectors or tensors. Specifically, for a connection on a fiber bundle, the parallel transport along every closed loop based at a fixed point yields a linear transformation of the fiber at that point. The collection of all such transformations forms the holonomy group of the connection at that point.

In a fiber bundle with a fixed connection, the holonomy group can depend on the chosen base point, but for a connected bundle the holonomy groups at different base points are conjugate within the structure group. The notion extends naturally to principal bundles and to vector bundles equipped with a connection.

In Riemannian geometry, the Levi-Civita connection yields a holonomy group Hol_p(M,g) contained in the orthogonal group of the tangent space at p. If the manifold is oriented, Hol_p lies in the special orthogonal group. For connected manifolds, the holonomy groups at different points are conjugate, so the holonomy is an intrinsic feature of the geometry.

The Ambrose–Singer theorem describes the Lie algebra of the holonomy group: it is generated by curvature endomorphisms R(X,Y) of the tangent space, transported to p. Intuitively, curvature accounts for the infinitesimal failure to return a vector to its original position after transporting along infinitesimal loops, and integrates to holonomy around finite loops.

Examples and significance: on flat spaces the holonomy is trivial; on the standard sphere the holonomy at a point is isomorphic to SO(2). Manifolds with special holonomy, such as SU(n) (Calabi–Yau), Sp(n) (hyperkähler), G2, or Spin(7), have rich geometric and physical implications.

Holonomy has applications in geometry, topology, and physics, including gauge theory and general relativity, where it encodes how curvature influences parallel transport around loops.