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We consider extension of the Einstein-Hilbert Lagrangian to a
general function of metric and Riemann curvature tensor, ${\cal
L}(g_{\mu\nu}, R^\mu_{\ \ \alpha\beta\nu})$. In these extension the
metric and the connection could be thought as independent degrees of
freedom (the Palatini formulation) or the connection could be
replaced by Levi-Civita connection expressed in terms of derivative
of metric (the metric formulation). For a general $\cL$ the Palatini
and metric formulations lead to different dynamical equations for
gravity. In this letter we show that requiring the equivalence of
the two formulations, at the level of equations of motion, severely
restricts the possible forms of the Lagrangians. We show that the
Lovelock gravity theories are the only theories of modified gravity
which satisfy this criterion.
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