Field equations for gravity quadratic in the curvature

Vacuum field equations for gravity are studied having their origin in a Lagrangian quadratic in the curvature. The motivation for this choice of the Lagrangian—namely the treating of gravity in a strict analogy to gauge theories of Yang-Mills type—is criticized, especially the implied view of connections as gauge potentials with no dynamical relation to the metric. The correct field equations with respect to variation of the connections and the metric independently are given. We deduce field equations which differ from previous ones by variation of the metric, the torsion, and the nonmetricity from which the connections are built.