Parallel Translations for a Left Invariant Spray

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 parallel translations, we provide alternative interpretations or proofs for some homogeneous curvature formulae. In particular, the Riemannian curvture appears in both a double Lie derivative along the spray vector and brackets between smooth vector fields induced by the connection operator. 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.