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