Parallel translations for a left invariant spray

TL;DR

This paper investigates the geometry of left invariant sprays on Lie groups, deriving differential equations for parallel translations and exploring their implications for curvature and holonomy, with connections to Finsler geometry.

Contribution

It introduces differential equations for parallel translations in left invariant spray structures using invariant frames, and proposes new questions related to Landsberg problem and holonomy.

Findings

Derived equations for linear and nonlinear parallel translations.

Provided alternative proofs for homogeneous curvature formulas.

Posed new questions on Landsberg problem and holonomy in spray geometry.

Abstract

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.