These conditions can be simplified in the case where X is symmetric space.

The dimension of is called the rank of the Hermitian symmetric space.

The holonomy was introduced by in order to study and classify symmetric spaces.

This suggests that the spatial separation of nearby points in a spherically symmetric space is.

When s is independent of X, M is a symmetric space.

Thus the Hermitian symmetric spaces are easily read off of the classification.

Thus the quaternion-Kähler symmetric spaces are easily read off from the classification.

The restrcted root system of a symmetric space and its dual can be identified.

In symmetric 3-dimensional space this exchange results in the inverse square law for electric force.

It is a special case of a more general result for symmetric spaces.