On a Riemannian manifold , tangent vectors can be moved along a path by parallel transport, which preserves vector addition and scalar multiplication. So a closed loop at a base point , gives rise to a invertible linear map of , the tangent vectors at . It is possible to compose closed loops by following one after the other, and to invert them by going backwards. Hence, the set of linear transformations arising from parallel transport along closed loops is a group, called the holonomy group.
Since parallel transport preserves the Riemannian metric, the holonomy group is contained in the orthogonal group . Moreover, if the manifold is orientable, then it is contained in the special orthogonal group. A generic Riemannian metric on an orientable manifold has holonomy group , but for some special metrics it can be a subgroup, in which case the manifold is said to have special holonomy.
A Kähler manifold is a -dimensional manifold whose holonomy lies in the unitary group . A Calabi-Yau manifold is a simply connected -dimensional manifold with holonomy in the special unitary group. A -dimensional manifold with holonomy group , the quaternionic unitary group, is called a hyper-Kähler manifold, and one with holonomy is called a quaternion Kähler manifold. The possible groups that can arise as a holonomy group of the metric compatible Levi-Civita connection were classified by Berger. The other possibilities for a nonproduct, nonsymmetric manifold are the Lie groups and . (Note that while Berger (1955) listed as a possibility of a Riemannian nonsymmetric, holonomy group, this possibility was excluded by Gray and Brown (1972).)
On a flat manifold, two homotopic loops give the same linear transformation. Consequently, the holonomy group is a group representation of the fundamental group of . In general though, the curvature of changes the parallel transport between homotopic loops. In fact, there is a formula for the difference as an integral of the curvature.