holonomy
Sign in to savealt=Visualisation of parallel transport on a sphere|thumb|Parallel transport on a sphere along a piecewise smooth path. The initial vector is labelled as V, parallel transported along the curve, and the resulting vector is labelled as \mathcal{P}_{\gamma}(V). The outcome of parallel transport will be different if the path is varied.
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Article
20 sectionsContents
- Definitions
- Holonomy of a connection in a vector bundle
- Holonomy of a connection in a principal bundle
- Holonomy bundles
- Monodromy
- Local and infinitesimal holonomy
- Ambrose–Singer theorem
- Riemannian holonomy
- Reducible holonomy and the de Rham decomposition
- The Berger classification
- Special holonomy and spinors
- Applications
- String theory
- Machine learning
- Affine holonomy
- Etymology
- See also
- Notes
- References
- Further reading
alt=Visualisation of parallel transport on a sphere|thumb|Parallel transport on a sphere along a piecewise smooth path. The initial vector is labelled as V, parallel transported along the curve, and the resulting vector is labelled as \mathcal{P}_{\gamma}(V). The outcome of parallel transport will be different if the path is varied.
In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence of the curvature of the connection. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.