Conjugate covariant derivatives on vector bundles and duality

Abstract

The notion of {\it conjugate connections}, discussed in \cite{be:c} for a given manifold $M$ and its tangent bundle, is extended here to covariant derivatives on an arbitrary vector bundle $E$ endowed with quadratic endomorphisms. The main property of pairs of such covariant derivatives, namely the duality, is pointed out. As generalization, the case of anchored (particularly Lie algebroid) covariant derivatives on $E$ is considered. As applications we study the Finsler bundle of $M$ as well as the Finsler connections on the slit tangent bundle of a Finsler geometry.