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LEFT INVARIANT CLIFFORD-WOLF HOMOGENEOUS (α, β)-METRICS ON COMPACT SEMISIMPLE LIE GROUPS

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Abstract

Let (M, F) be a connected Finsler space. An isometry of (M, F) is called a Clifford-Wolf translation (or simply CW-translation) if it moves all points the same distance. The compact Finsler space (M, F) is called restrictively Clifford-Wolf homogeneous (restrictively CW-homogeneous) if for any point x ∈ M , there exists an open neighborhood U of x such that for any two x 1, x 2 ∈ U, there exists a CW-translation σ of (M, F) such that σ(x 1) = x 2. In this paper, we define the notion of a good normalized datum for a homogeneous non-Riemannian (α, β)-space, and use that to study the restrictive CW-homogeneity of left invariant (α, β)-metrics on a compact connected semisimple Lie group. We prove that a left invariant restrictively CW-homogeneous (α, β)-metric on a compact semisimple Lie group must be of the Randers type. This gives a complete classification of left invariant restrictively CW-homogeneous (α, β)-metrics on compact semisimple Lie groups.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    SHAOQIANG DENG

  2. College of Mathematics, Tianjin Normal University, Tianjin, 300387, People’s Republic of China

    MING XU

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  1. SHAOQIANG DENG
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  2. MING XU
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DENG, S., XU, M. LEFT INVARIANT CLIFFORD-WOLF HOMOGENEOUS (α, β)-METRICS ON COMPACT SEMISIMPLE LIE GROUPS. Transformation Groups 20, 395–416 (2015). https://doi.org/10.1007/s00031-014-9294-5

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