Compact homogeneous Riemannian manifolds with low co-index of symmetry

TL;DR

This paper develops a structure theory for compact homogeneous Riemannian manifolds based on the co-index of symmetry and classifies those with low co-index, providing new examples from symmetric space theory.

Contribution

It introduces a general structure framework for these manifolds and classifies irreducible, simply connected cases with co-index ≤ 3, expanding understanding of their geometry.

Findings

Classification of manifolds with co-index ≤ 3

Construction of new examples from symmetric space theory

Development of a general structure theory for these manifolds

Abstract

We develop a general structure theory for compact homogeneous Riemannian manifolds in relation to the co-index of symmetry. We will then use these results to classify irreducible, simply connected, compact homogeneous Riemannian manifolds whose co-index of symmetry is less or equal than three. We will also construct many examples which arise from the theory of polars and centrioles in Riemannian symmetric spaces of compact type.