  As is known from studies of gravity models in the Palatini formalism, there
exist two inequivalent definitions of the generalized Ricci tensor in terms of
the generalized curvature namely, $\widetilde{R}_{\mu\nu}=R^\rho_{\mu\rho\nu}$
and $R^\mu_{\nu}=g^{\alpha\beta}R^\mu_{\alpha\nu\beta}$. A deep formal
investigation of theories with lagrangians of the form
$L=L(\widetilde{R}_{(\mu\nu)})$ was initiated in [4]. In that work, the authors
leave the connection free, and find out that the torsion only appears as a
projective mode. This agrees with the widely employed condition of vanishing
torsion in these theories as a simple gauge choice. In the present work the
complementary scenario is studied namely, the one described by a lagrangian
that depends on the other possible Ricci tensor $L=L(R_{(\mu\nu)})$. The
torsion is completely characterized in terms of the metric and the connection,
and a rather detailed description of the equations of motion is presented. It
is shown that these theories are non trivial even for $1+1$ space time
dimensions, and admit non zero torsion even in this apparently simple case. It
is suggested that to impose zero torsion by force may result into an
incompatible system. In other words, the presence of torsion may be beneficial
for insuring that the equations of motion are well posed. The results presented
here do not contradict the results of [4], as the resulting theories are
inequivalent.
