Universal gravitational equations

We show that for a wide class of Lagrangians which depend only on the scalar curvature of a metric and a connection, the application of the first-order formalism, i.e. treating the metric and the connection as independent variables, leads to «universal» equations. If the dimensionn of space-time is greater than two, these universal equations are Einstein equations for a generic Lagrangian. There are exceptional cases where a bifurcation appears. In particular, bifurcations take place for conformally invariant Lagrangians $L = R^(n/2)√g$. For 2-dimensional space-time we obtain that the universal equation is 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 vector field.