Curvatures of homogeneous Randers spaces |
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Authors: | Shaoqiang Deng Zhiguang Hu |
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Institution: | 1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China;2. College of Mathematics, Tianjin Normal University, Tianjin 300387, People’s Republic of China |
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Abstract: | We study curvatures of homogeneous Randers spaces. After deducing the coordinate-free formulas of the flag curvature and Ricci scalar of homogeneous Randers spaces, we give several applications. We first present a direct proof of the fact that a homogeneous Randers space is Ricci quadratic if and only if it is a Berwald space. We then prove that any left invariant Randers metric on a non-commutative nilpotent Lie group must have three flags whose flag curvature is positive, negative and zero, respectively. This generalizes a result of J.A. Wolf on Riemannian metrics. We prove a conjecture of J. Milnor on the characterization of central elements of a real Lie algebra, in a more generalized sense. Finally, we study homogeneous Finsler spaces of positive flag curvature and particularly prove that the only compact connected simply connected Lie group admitting a left invariant Finsler metric with positive flag curvature is SU(2). |
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Keywords: | 22E46 53C30 53C35 53C60 |
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