Geodesic
Left Invariant Spray Structure on a Lie Group
Journal of Lie theory, Tome 32 (2022) no. 1, pp. 121-138
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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.
Classification : 53B40, 53C30, 53C60
Mots-clés : Finsler metric, Landsberg Conjecture, left invariant frame, Lie group, Riemann curvature, S-curvature, spray structure 
@article{JLT_2022_32_1_JLT_2022_32_1_a5,
     author = {M. Xu},
     title = {Left {Invariant} {Spray} {Structure} on a {Lie} {Group}},
     journal = {Journal of Lie theory},
     pages = {121--138},
     year = {2022},
     volume = {32},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JLT_2022_32_1_JLT_2022_32_1_a5/}
}
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M. Xu. Left Invariant Spray Structure on a Lie Group. Journal of Lie theory, Tome 32 (2022) no. 1, pp. 121-138. http://geodesic.mathdoc.fr/item/JLT_2022_32_1_JLT_2022_32_1_a5/