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 {pipe} · {pipe} 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 {pipe} · {pipe} 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. © 2014 Allerton Press, Inc.