Universality of the Einstein theory of gravitation

We show that generalizations of general relativity theory, which consist in replacing the Hilbert Lagrangian $L_{Hilbert}=\frac{1}{16\pi }\sqrt{\mathrm{\mid }g\mathrm{\mid }}R$ by a generic scalar density $L=L(g_{\mu \nu },R_{\mu \nu \kappa }^{\lambda })$ depending upon the metric $g_{\mu \nu }$ and the curvature tensor $R_{\mu \nu \kappa }^{\lambda }$, are equivalent to the conventional Einstein theory for a (possibly) different metric tensor ${\tilde{g}}_{\mu \nu }$ 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.