Editing Holonomy (section)

{{Use American English|date = March 2019}}
{{Short description|Concept in differential geometry}}
[[File:Holonomy-vis-with-label.png|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 <math>V</math>, parallel transported along the curve, and the resulting vector is labelled as <math>\mathcal{P}_{\gamma}(V)</math>. The outcome of parallel transport will be different if the path is varied.]]

{{Technical|date=October 2024}}

In [[differential geometry]], the '''holonomy''' of a [[connection (mathematics)|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.

Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy.  The most common forms of holonomy are for connections possessing some kind of [[symmetry]].  Important examples include: holonomy of the [[Levi-Civita connection]] in [[Riemannian geometry]] (called '''Riemannian holonomy'''), holonomy of [[connection (vector bundle)|connections]] in [[vector bundle]]s, holonomy of [[Cartan connection]]s, and holonomy of [[connection (principal bundle)|connections]] in [[principal bundle]]s.  In each of these cases, the holonomy of the connection can be identified with a [[Lie group]], the '''holonomy group'''.  The holonomy of a connection is closely related to the curvature of the connection, via the ''[[#Ambrose–Singer_theorem|Ambrose–Singer theorem]]''.

The study of Riemannian holonomy has led to a number of important developments.  Holonomy was introduced by {{harvs|txt|last=Cartan|first=Élie|authorlink = Élie Cartan|year=1926}} in order to study and classify [[symmetric space]]s.  It was not until much later that holonomy groups would be used to study Riemannian geometry in a more general setting. In 1952 [[Georges de Rham]] proved the ''de Rham decomposition theorem'', a principle for splitting a Riemannian manifold into a [[Cartesian product]] of Riemannian manifolds by splitting the [[tangent bundle]] into irreducible spaces under the [[group action|action]] of the local holonomy groups.  Later, in 1953, [[Marcel Berger]] classified the possible irreducible holonomies.  The decomposition and classification of Riemannian holonomy has applications to physics and to [[string theory]].