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Journal of Mathematical Sciences

, Volume 161, Issue 1, pp 97–121 | Cite as

The Chebyshev norm on the lie algebra of the motion group of a compact homogeneous Finsler manifold

  • V. N. Berestovskii
  • Yu. G. Nikonorov
Article

Abstract

In this paper, we prove that the natural metric on the connected component of the unit in the (Lie) motion group of a compact Finsler manifold supplied with its inner metric generates a bi-invariant inner Finsler metric. The latter is defined by the invariant Chebyshev norm on the Lie algebra of generators of 1-parameter motion subgroups on the manifold. This norm is equal to the maximal value of the generator’s length. A δ-homogeneous manifold is characterized by the condition that the canonical projection of the component onto the manifold is a submetry with respect to their inner metrics. The Chebyshev norms for the Euclidean spheres, the Berger spheres, and homogeneous Riemannian metrics on the 3-dimensional complex projective space are found. This gives interesting examples of invariant norms on Lie algebras and a new method for the separating of delta-homogeneous but not normal metrics.

Keywords

Riemannian Manifold Weyl Group Cartan Subalgebra Invariant Norm Riemannian Submersion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Omsk Branch of Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesOmskRussia
  2. 2.Rubtsovsk Industrial InstituteAltai State Technical UniversityRubtsovskRussia

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