  The aim of the present paper is to provide an \emph{intrinsic} investigation
of the properties of the most important geometric objects associated with the
fundamental linear connections in Finsler geometry. We investigate
intrinsically the most general relations concerning the torsion tensor fields
and the curvature tensor fields associated with a given regular connection on
the pullback bundle of a Finsler manifold. These relations, in turn, play a key
role in obtaining other interesting results concerning the properties of the
most important geometric objects associated with the fundamental canonical
linear connections on the pullback bundle of a Finsler manifold, namely, the
Cartan connection, the Berwald connection, the Chern (Rund) connection and the
Hashiguchi connection.
  For the sake of completeness and for comparison reasons, we provide an
appendix presenting a global survey of canonical linear connections in Finsler
geometry and the fundamental geometric objects associated with them.
