Holonomy group of Riemannian metric

Definition

Let $M$ be a differential manifold and $g$ a Riemannian metric on $M$ (turning $(M,g)$ into a Riemannian manifold). The holonomy group of $g$ is defined as the holonomy group for the Levi-Civita connection on the tangent bundle.

The holonomy group is a subgroup of the orthogonal group. This is because transport along a curve using the Levi-Civita connection (that is, Levi-Civita transport) preserves the Riemannian metric.

Related notions

• Holonomy group
• Restricted holonomy group of Riemannian metric
• Riemannian metric with special holonomy

Facts

Orientedness

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.

Flat manifolds

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.