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Homogeneous almost normal Riemannian manifolds


In this article, we introduce a newclass of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive Lie group G of isometries for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν such that the canonical projection p: (G, ν) (G/H, µ) is a Riemannian submersion and the norm | · | of the product ν is at least the bi-invariant Chebyshev normon G defined by the space (M,µ).We prove the following results: Every homogeneous Riemannian manifold is almost normal homogeneous. Every homogeneous almost normal Riemannian manifold is naturally reductive and generalized normal homogeneous. For a homogeneous G-normal Riemannian manifold with simple Lie group G, the unit ball of the norm | · | is a Löwner-John ellipsoid with respect to the unit ball of the Chebyshev norm; an analogous assertion holds for the restrictions of these norms to a Cartan subgroup of the Lie group G. Some unsolved problems are posed.

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Correspondence to V. N. Berestovskiĭ.

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Original Russian Text © V.N. Berestovskiĭ, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 1, pp. 18–27.

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Berestovskiĭ, V.N. Homogeneous almost normal Riemannian manifolds. Sib. Adv. Math. 24, 12–17 (2014).

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  • Weyl group
  • naturally reductive Riemannian manifold
  • Chebyshev norm
  • homogeneous normal Riemannian manifold
  • homogeneous generalized normal Riemannian manifold, homogeneous almost normal Riemannian manifold
  • Cartan subagebra
  • Löwner-John ellipsoid