Metric structure of symmetric spaces

Abstract

For each symmetric space, there is a corresponding pair of Lie groups called the symmetric pair. This pair describes the geometry of the symmetric space by its Lie structure. Some symmetric spaces can be presented by Lie groups different from the symmetric pair, but unfortunately with this nonstandard presentation the geometry of the symmetric space cannot be easily described as the case of a symmetric pair. In this paper, we consider a symmetric space presented by a nonsymmetric pair, and show that some geometric properties are still described by its Lie structure, and in particular we calculate the curvature in this case.