On the direction independence of two remarkable Finsler tensors

On the projective theory of sprays with applications to Finsler geometry

“…Briefly, we apply the pull-back formalism. Thus, for example, we re-prove our earlier result published in [8], where we used classical tensor calculus. The self-containedness of our exposition also means that we prove most of the classically well-known auxiliary results whose formulation and proof is not available (in our best knowledge) in our context.…”

“…since P is homogeneous of degree 0. Now we verify by an index-free argument the following result of our above mentioned paper [8]: Proposition 7.7 If the stretch tensor of a Finsler manifold depends only on the position, then it vanishes identically, i.e., ∇ v Σ = 0 implies that Σ = 0.…”

P-Berwald manifolds

“…For the readers' convenience, we briefly summarize these basic technicalities in Section 2, and, partly, in Section 3. We follow the notation and conventions of reference [17] and, with some modifications, [5]. These papers also give some links to the classical approach.…”