Homogeneous almost normal Riemannian manifolds

In this article, we introduce a newclass of compact homogeneous Riemannian manifolds $(M=G/H,\mu)$ 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 $\nu$ such that the canonical projection $p\colon(G,\nu)\rightarrow(G/H,\mu)$ is a Riemannian submersion and the norm ${|\boldsymbol\cdot|}$ of the product $\nu$ is at least the bi-invariant Chebyshev normon $G$ defined by the space $(M,\mu)$. 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 ${|\boldsymbol\cdot|}$ 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.