I) In Palatini $f(R)$ gravity, the Lagrangian density is

$$ {\cal L}~=~ \sqrt{-g} f(R), $$

with $$R~:=~ g^{\mu\nu} R_{\mu\nu}(\Gamma),$$

and where $\Gamma^{\lambda}_{\mu\nu}=\Gamma^{\lambda}_{\nu\mu}$ is an arbitrary torsionfree$^1$ connection.

II) As OP mentions, the word *Palatini* refers to that the metric $g_{\mu\nu}$ and the connection $\Gamma^{\lambda}_{\mu\nu}$ are *independent* variables. We therefore get two types of EL equations:

The EL equations for the metric $g_{\mu\nu}$ are the generalization of EFE.

The EL equations for the connection $\Gamma^{\lambda}_{\mu\nu}$ turns out to be the metric compatibility condition for a second metric defined as
$$ \hat{g}_{\mu\nu}~:=~f^{\prime}(R) g_{\mu\nu}. $$
In other words, the classical solution for $\Gamma^{\lambda}_{\mu\nu}$ is the Levi-Civita connection for the second metric $\hat{g}_{\mu\nu}$.

III) So Einstein gravity (GR) with a possible cosmological constant

$$ f(R)~=~R-2\Lambda, $$

or equivalently

$$ f^{\prime}(R)~=~1,$$

corresponds to the special case where the two metrics $g_{\mu\nu}$ and $\hat{g}_{\mu\nu}$ coincide, and hence $\Gamma^{\lambda}_{\mu\nu}$ becomes the Levi-Civita connection for $g_{\mu\nu}$.

--

$^1$ One could allow a non-dynamical torsion piece, but we will not pursuit this here for simplicity. For more on torsion, see e.g. also this Phys.SE post.

This post imported from StackExchange Physics at 2014-10-23 07:31 (UTC), posted by SE-user Qmechanic