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

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.