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 Mat. Tr., 2013, Volume 16, Number 1, Pages 18–27 (Mi mt247)

Homogeneous almost normal Riemannian manifolds

V. N. Berestovskiĭ

Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Omsk, Russia

Abstract: 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.

Key words: 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.

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English version:
Siberian Advances in Mathematics, 2014, 24:1, 12–17

Bibliographic databases:

UDC: 514.70

Citation: V. N. Berestovskiǐ, “Homogeneous almost normal Riemannian manifolds”, Mat. Tr., 16:1 (2013), 18–27; Siberian Adv. Math., 24:1 (2014), 12–17

Citation in format AMSBIB
\Bibitem{Ber13} \by V.~N.~Berestovski{\v\i} \paper Homogeneous almost normal Riemannian manifolds \jour Mat. Tr. \yr 2013 \vol 16 \issue 1 \pages 18--27 \mathnet{http://mi.mathnet.ru/mt247} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3156671} \transl \jour Siberian Adv. Math. \yr 2014 \vol 24 \issue 1 \pages 12--17 \crossref{https://doi.org/10.3103/S1055134414010027}