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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 space-time is greater than two these universal equations are Einstein equations for a generic Lagrangian and are suitably replaced by other universal equations at bifurcation points. We show that bifurcations take place in particular for conformally invariant Lagrangians L = R n/2 √ g and prove that their solutions are conformally equivalent to solutions of Einstein equations. For 2-dimensional space-time we find instead that the universal equation is always the equation of constant scalar curvature; the connection in this case is a Weyl connection, containing the Levi-Civita connection of the metric and an additional vectorfield ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their bifurcations.

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It has been recently shown that, in the rst order (Palatini) formalism, there is universality of Einstein equations and Komar energy{ momentum complex, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets Einstein equations and Komar's expression for the energy{momentum complex. In this paper a similar analysis (also in the framework of the rst order formalism) is performed for all nonlinear Lagrangians depending on the (symmetrized) Ricci square invariant. The main result is that the universality of Einstein equations and Komar energy{momentum complex also extends to this case (modulo a conformal transformation of the metric).

Abstract.We criticize and generalize some properties of Nöther charges presented in a paper by V. Iyer and R. M. Wald and their application to entropy of black holes. The first law of black holes thermodynamics is proven for any gauge-natural field theory. As an application charged Kerr-Newman solutions are considered. As a further example we consider a (1 + 2) black hole solution.

Among the so-called 'non-linear' (purely metric) Lagrangians for the gravitational field, those which depend in a quadratic way on the components of the Riemann tensor have been given particular consideration by many authors. In this paper, the authors deal with the most general quadratic Lagrangian depending on the full Riemann tensor, in arbitrary dimension; instead of considering the corresponding fourth-order Euler-Lagrange equations, they investigate an equivalent set of second-order quasilinear equations which are obtained by (a suitably generalised) Legendre transformation. In this framework, they compare this class of theories with those depending on the Ricci tensor only, showing that the Weyl tensor dependence breaks the equivalence with general relativity, but the new auxiliary field arising in this case has no dynamical term. The degeneracy occurring for a suitable choice of the parameters in the Lagrangian is widely discussed, and some effects of a non-minimal coupling with an external scalar field are also described.