The Chebyshev norm on the lie algebra of the motion group of a compact homogeneous Finsler manifold |
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Authors: | V N Berestovskii Yu G Nikonorov |
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Institution: | (1) Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Omsk, Russia;(2) Rubtsovsk Industrial Institute, Altai State Technical University, Rubtsovsk, Russia |
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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.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra,
2008. |
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