Holonomy group of Riemannian metric
Let be a differential manifold and a Riemannian metric on (turning into a Riemannian manifold). The holonomy group of is defined as the holonomy group for the Levi-Civita connection on the tangent bundle.
- Holonomy group
- Restricted holonomy group of Riemannian metric
- Riemannian metric with special holonomy
If the Riemannian manifold is oriented, the holonomy group is a subgroup of the special orthogonal group.
Generic and special holonomy
For a generic Riemannian metric on an oriented manifold, the holonomy group is the whole special orthogonal group. A Riemannian metric where the holonomy group is a proper subgroup of the special orthogonal group is termed a Riemannian metric with special holonomy.
For a flat metric, the restricted holonomy group is trivial, or in other words, the holonomy of any loop homotopic to the identity, is trivial. Thus, the holonomy group is a homomorphic image of the fundamental group, and thus gives rise to a linear representation of the fundamental group.