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

Compact homogeneous Riemannian manifolds with low co-index of symmetry