On the Bach and Einstein equations in presence of a field

The aim of this paper is to introduce and justify a possible generalization of the classic Bach field equations on a four dimensional smooth manifold $M$ in presence of field $\varphi$, that in this context is given by a smooth map with source $M$ and target another Riemannian manifold. Those equations are characterized by the vanishing of a two times covariant, symmetric, traceless and conformally invariant tensor field, called $\varphi$-Bach tensor, that in absence of the field $\varphi$ reduces to the classic Bach tensor. We provide a variational characterization for $\varphi$-Bach flat manifolds and we do the same also for harmonic-Einstein manifolds, i.e., solutions of the Einstein field equations with source the conservative field $\varphi$. We take the opportunity to discuss a generalization of some related topics: the Yamabe problem, the image of the scalar curvature map, warped product solutions and static manifolds.