  We study models where a scalar field has derivative and non-derivative
couplings to the Ricci tensor and the co-Ricci tensor with a view to inflation.
We consider both the metric formulation and the Palatini formulation. In the
Palatini case, the couplings to the Ricci tensor and the Ricci scalar give the
same result regardless of whether the connection is unconstrained or the
non-metricity or the torsion is assumed to vanish. When the co-Ricci tensor is
included, the unconstrained case and the zero torsion case are physically
different. We reduce all the actions to the Einstein frame with minimally
coupled matter, and find the leading order differences between the metric case
and the Palatini cases.
