  In Eddington gravity, the action principle involves only the symmetric parts
of the connection and the Ricci tensor, with a metric that emerges
proportionally to the latter. Here, we relax this symmetric character, prolong
the action with the antisymmetric parts of the Ricci term, and allow for
various couplings with scalar fields. We propose two possible invariant actions
formed by distinct combinations of the independent Ricci tensors and show that
the generated metric must involve an additional antisymmetric part due to the
relaxation of the symmetrization property. The comprehensive study shows that
the second curvature influences the dynamics of the connection, hence its
solution in terms of the metric, and the evolution of the scalar fields. These
new dynamical features are expected to stand viable and to have interesting
implications in domains where scalar fields are indispensable.
