Parallel translations for a left invariant spray

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.

Submersion and homogeneous spray geometry

“…In this section, we recall the technique of global invariant frames on a Lie group and collect some known results for a left invariant spray structure from [39,40].…”

“…Let {∂ x i , ∂ y i , ∀i} be the frame for a standard local coordinate (x i , y j ) on G. Then the transformation between {U i , ∂ u i , ∀i} and {∂ x i , ∂ y i , ∀i} is the following [40],…”

“…The other hint is from the recent progress in the study of a left invariant spray structure G on a Lie group G [39,40]. Here the Lie group G is viewed as the coset space G/H = G/{e} with the unique linear decomposition g = h + m = 0 + g, which is obviously reductive.…”

“…By left translations, geodesics and parallel translations along smooth curves on (G, G) are corresponded to integral curves of −η and some ordinary differential equations (i.e., ODEs in short) on T e G = g respectively. See [39,40] or Section 4 for the precise statements. To summarize, Question 1.1 can be answered for a left invariant spray structure, using the technique of global invariant frames.…”

“…Notice that the second statement in Theorem F was firstly proved in [40] for a left invariant spray structure, where a slight generalization for the bracket between smooth vector fields is needed (see Remark 1.3 in [40] or Remark 6.11).…”