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 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.