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