Holonomy Group

The holonomy group of a Riemannian manifold consists of the linear transformations obtained by parallel transport along closed loops at a basepoint . Parallel transport of manifold tangent vectors along a path preserves vector addition and scalar multiplication. So a loop at the basepoint gives rise to an invertible linear map of , the tangent space 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.

The restricted holonomy group is obtained by considering only loops that are homotopic to the constant loop at . It is the connected component of the identity element in the full 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 Riemannian metrics it can be a proper subgroup of , 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 smooth Calabi-Yau space is a -dimensional manifold with holonomy in the special unitary group. A -dimensional manifold with holonomy group contained in , the quaternionic unitary group, is called a hyper-Kähler manifold, and, for , one with holonomy is called a quaternion Kähler manifold. Berger (1955) classified the possible restricted holonomy groups of the metric-compatible Levi-Civita connection on locally irreducible, locally nonsymmetric Riemannian manifolds. Under these hypotheses, the other possibilities are the Lie groups and . Berger (1955) also listed as a possible restricted holonomy group under these hypotheses, but Brown and Gray (1972) excluded this possibility.

On a flat manifold, two homotopic loops give the same linear transformation. Consequently, parallel transport defines a group representation of the fundamental group of whose image is the holonomy group. In general though, the Riemann tensor of affects the parallel transport between homotopic loops. In fact, there is a formula for the difference as an integral of the Riemann tensor.