# Holonomy group of Riemannian metric

Template:Riemannian metric-associated group

## Contents

## Definition

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.

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.