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Universality of the Einstein theory of gravitation

We show that generalizations of general relativity theory, which consist in replacing the Hilbert Lagrangian $L_{Hilbert} = \frac 1{16π} \sqrt{|g|} R$ by a generic scalar density $L=L(g_{μν}, R^λ_{μνκ})$ depending upon the metric $g_{μν}$ and the curvature tensor $R^λ_{μνκ}$, are equivalent to the conventional Einstein theory for a (possibly) different metric tensor $\tilde{g}_{μν}$ and (possibly) a different set of matter fields. The simple proof of this theorem relies on a new approach to variational problems containing metric and connection.