Homogeneous almost normal Riemannian manifolds

被引:0
|
作者
Berestovskiĭ V.N. [1 ]
机构
[1] Sobolev Institute of Mathematics, Omsk Division, Omsk
基金
俄罗斯基础研究基金会;
关键词
Cartan subagebra; Chebyshev norm; homogeneous generalized normal Riemannian manifold; homogeneous almost normal Riemannian manifold; homogeneous normal Riemannian manifold; Löwner-John ellipsoid; naturally reductive Riemannian manifold; Weyl group;
D O I
10.3103/S1055134414010027
中图分类号
学科分类号
摘要
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.
引用
收藏
页码:12 / 17
页数:5
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