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Journal of Lie Theory 32 (2022), No. 1, 121--138
Copyright Heldermann Verlag 2022

Left Invariant Spray Structure on a Lie Group

Ming Xu
School of Mathematical Sciences, Capital Normal University, Beijing 100048, P. R. China

We use the technique of invariant frame to study the left invariant spray structure on a Lie group. We calculate its S-curvature and Riemann curvature, which generalizes L. Huang's formulae in homogeneous Finsler geometry. Using the canonical bi-invariant spray structure as the origin, any left invariant spray structure can be associated with a spray vector field on the Lie algebra. We find the correspondence between the geodesics for a left invariant spray structure and the inverse integral curves of its spray vector field. As an application for this correspondence, we provide an alternative proof of Landsberg Conjecture for homogeneous Finsler surfaces.

Keywords: Finsler metric, Landsberg Conjecture, left invariant frame, Lie group, Riemann curvature, S-curvature, spray structure.

MSC: 53B40, 53C30, 53C60.

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