In differential geometry, the holomy of a smooth variety is a general geometric result of the curvature of the joint, which the degree in which parallel transport around closed loops fails, the geometric data being transported, to save. For flat connections, the associated holonomy is a kind of monodromy, and holonomy is an inherently global concept. For curved connections, holonomy has non-trivial local and global features. According to Ambrose-Singer's statement, the holonomy of a connection is closely linked to the curve.
Holonomy is most common in compounds that have a certain degree of symmetry. An example of such a connection from the Riemann geometry is the Levi-Civita connection. Other examples are compounds in vector bundles, the holonomy in a Cartan compound and the holonomy of compounds in main beads. In all these cases, the holonomy of the connection can be described as a Lie group.
The concept of holonomy was first used in 1926 by Élie Cartan to study and describe symmetrical spaces. Also see
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