
Journal of Lie Theory 32 (2022), No. 1, 121138 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 mgmgmgxu@163.com We use the technique of invariant frame to study the left invariant spray structure on a Lie group. We calculate its Scurvature and Riemann curvature, which generalizes L. Huang's formulae in homogeneous Finsler geometry. Using the canonical biinvariant 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, Scurvature, spray structure. MSC: 53B40, 53C30, 53C60. [ Fulltextpdf (165 KB)] for subscribers only. 