On parallel transport in Finsler spaces

I discuss the properties of parallel transport — that is, the generally nonlinear parallel transport defined by the Ehresmann connection, or horizontal distribution — in Finsler spaces. I explain its relationship to the Berwald (linear) connection. I derive a number of criteria in terms of parallel transport for a Finsler space to have some special property: starting of course with <span class="small-caps">Ichijyō</span>'s long-established result that a Finsler space is a Landsberg space if and only if parallel transport is always an isometry of the fibre metric ([9], [10]); deriving conditions for a Finsler space to be Berwald, weakly Landsberg, or weakly Berwald; and ending with the new result that a Finsler space is a Landsberg space if and only if parallel transport is always a projective transformation of the Levi-Civita connection of the fibre metric.