  Introducing the Lie algebroid generalized tangent bundle of a Kaluza-Klein
bundle, we develop the theory of general distinguished linear connections for
this space. In particular, using the Lie algebroid generalized tangent bundle
of the Kaluza-Klein vector bundle, we present the $\left( g,h\right) $-lift of
a curve on the base $M$ and we characterize the horizontal and vertical
parallelism of the $\left( g,h\right) $-lift of accelerations with respect to a
distinguished linear $\left( \rho ,\eta \right) $-connection. Moreover, we
study the torsion, curvature and Ricci tensor field associated to a
distinguished linear $\left( \rho ,\eta \right) $-connection and we obtain the
identities of Cartan and Bianchi type in the general framework of the Lie
algebroid generalized tangent bundle of a Kaluza-Klein bundle. Finally, we
introduce the theory of (pseudo) generalized Kaluza-Klein G-spaces and we
develop the Einstein equations in this general framework.
