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In differential geometry, the holonomy of a given structure (for example a Riemannian metric, or more general G-structure ) at a point P on a smooth manifold M is the group of all linear maps transforming the tangent space at P that can be induced by a parallel transport around a loop based at P. A generic d-dimensional Riemannian manifold has an O(d) holonomy, or SO(d), if it is orientable. A flat d-dimensional manifold (such as the torus) has as holonomy a discrete group.

The Berger list from 1955 was a list of holonomy groups possible in the case of an affine connection, which allowed for a finite number of further exceptional cases. Research from 1995 showed some modification was necessary.

Intermediate cases are the most interesting ones. Consider Calabi-Yau manifolds: SU(3) holonomy is the maximum holonomy of six-dimensional (three-complex-dimensional) Calabi-Yau manifolds. A generic six-dimensional manifoldIn mathematics, a manifold ''M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. Therefor has an O(6) holonomy (or SO(6), if it is orientable). A flat six-dimensional manifold (such as the six-torus) has a discrete holonomy. SU(3) is something in between; a subgroup of SU(4) (which is locally isomorphic to SO(6)) that guarantees that one quarter of the original supersymmetryIn particle physics, supersymmetry is a symmetry that relates bosons and fermions. In supersymmetric theories, every fundamental fermion has a superpartner which is a boson and vice versa. Although supersymmetry has yet to be observed in the real world it will be preserved if a manifold of SU(3) holonomy is used for compactification of string theoryA string theory is a physical model whose fundamental building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that were the basis of most earlier physics. For this reason, string theories are able. A generic Kähler manifoldIn mathematics, a Kahler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. would have a U(3) holonomy; and the component of the holonomy group which contains the identity is restricted to be in the SU(3) group if the manifold is Ricci flat.

Calabi-Yau d-folds have an SU(d) holonomy and they are a special case of Kähler manifolds with U(d) holonomy. There exist seven-dimensional real manifolds with holonomy G₂Lie groups In mathematics, G is the name of a Lie group and also its Lie algebra. It is the smallest of the five exceptional simple Lie groups. G has rank 2 and dimension 14. Its fundamental representation is 7-dimensional. G can be described as the autom - the so-called G2 manifoldsA G manifold also known as a Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group G''. The group is one the five exceptional simple Lie groups. It can be described as the automorphism group of the octonions, or equivalently, as a - as well as eight-dimensional real manifolds with holonomy , hyperkähler manifolds with holonomy — the real dimension must be a multiple of four in this case — and these hyperkähler manifolds are a special class of quaternion Kähler manifold s with holonomy locally isomorphic to .

The local holonomy group is the subgroup of the holonomy group that only contains the transformations induced by the parallel transport along contractible loops. Therefore, the notions of holonomy and local holonomy are equivalent for simply connected manifolds.