The Chebyshev norm on the lie algebra of the motion group of a compact homogeneous Finsler manifold
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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.
KeywordsRiemannian Manifold Weyl Group Cartan Subalgebra Invariant Norm Riemannian Submersion
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