The index of symmetry of compact naturally reductive spaces

TL;DR

This paper introduces the index of symmetry, a geometric invariant quantifying how close a Riemannian manifold is to being symmetric, and computes it for compact naturally reductive spaces, revealing new geometric insights.

Contribution

It defines the index of symmetry and provides a geometric method to compute it for compact naturally reductive spaces, including novel examples with non-group type leaves of symmetry.

Findings

The index of symmetry can be explicitly computed for compact naturally reductive spaces.

The leaf of symmetry is of group type in these spaces.

Examples with non-group type leaves of symmetry are identified.

Abstract

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.