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On flag curvature and homogeneous geodesics of left invariant Randers metrics on the semi-direct product \(\mathfrak{a}\oplus _p\mathfrak{r}\). (English) Zbl 1426.53084

Summary: We study flag curvature and homogeneous geodesics of left invariant Randers metrics on the Lie group with Lie algebra \(\mathfrak{a}\oplus_{\mathfrak{p}}\mathfrak{r}\), where \(\mathfrak{a}\) and \(\mathfrak{r}\) are abelian Lie algebra of dimension \(n\) and \(1\), respectively. We give their flag curvature formulas explicitly. We show that there is an \((n+1)-\) dimensional Lie group with left invariant Randers metric which admits exactly one homogeneous geodesic.

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C30 Differential geometry of homogeneous manifolds
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