The universality of vacuum Einstein equations with cosmological constant

It is shown that for a wide class of analytic Lagrangians, which depend only on the scalar curvature of a metric and a connection, the application of the so called `Palatini formalism', i.e. treating the metric and the connection as independent variables, leads to `universal' equations. If the dimension n of spacetime is greater than two these universal equations are vacuum Einstein equations with cosmological constant for a generic Lagrangian and are suitably replaced by other universal equations at degenerate points. We show that degeneracy takes place in particular for conformally invariant Lagrangians $L=R^(n/2)\sqrt{g}$ and we prove that their solutions are conformally equivalent to solutions of Einstein's equations. For two-dimensional spacetimes we find instead that the universal equation is always the equation of constant scalar curvature; in this case the connection is a Weyl connection, containing the Levi-Civita connection of the metric and an additional vector field ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their degenerate points.