On homogeneous Landsberg surfaces

Tayebi, A.; Najafi, Behzad

doi:10.1016/j.geomphys.2021.104314

Cited by 5 publications

(1 citation statement)

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“…Landsberg Problem asks if there exists a (regular) Landsberg metric which is not Berwald [12]. See [15][17] and the references therein for some recent progress on this problem. Theorem 1.5 implies that, if G is induced by a left invariant Landsberg metric, then H is a Lie subalgebra in the space of all Killing vector fields for the Hessian metric of F (e, •) on g\{0}, which must have a finite dimension.…”

Section: 3mentioning

confidence: 99%

Parallel translations for a left invariant spray

Xu¹

2021

Preprint

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In this paper, we study the left invariant spray geometry on a connected Lie group. Using the technique of invariant frames, we find the ordinary differential equations on the Lie algebra describing for a left invariant spray structure the linearly parallel translations along a geodesic and the nonlinearly parallel translations along a smooth curve. In these equations, the connection operator plays an important role. Using linearly parallel translations, we provide alternative interpretations or proofs for some homogeneous curvature formulae. Concerning the nonlinearly ones, we propose two questions in left invariant spray geometry. One question generalizes Landsberg Problem in Finsler geometry, and the other concerns the restricted holonomy group.

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Section: 3mentioning

confidence: 99%