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Curvatures of homogeneous Randers spaces. (English) Zbl 1281.53075

Summary: 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 \(\mathrm{SU}(2)\).

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
22E46 Semisimple Lie groups and their representations
53C30 Differential geometry of homogeneous manifolds
53C35 Differential geometry of symmetric spaces
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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