Holonomy group of Riemannian metric

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Template:Riemannian metric-associated group

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

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.