Birkhoff theorems forR+R2gravity theories with torsion

A Birkhoff theorem recently proved by Ramaswamy and Yasskin for one specific $R+R^2$ gravity theory with torsion is extended to include most $R+R^2$ theories. Out of the five-parameter family of $R+R^2$ theories, two four-parameter families are shown to obey a Birkhoff theorem. In addition, the entire five-parameter family is shown to obey a weakened Birkhoff theorem: If the assumption of asymptotic flatness is added to the assumption of spherical symmetry, then the only solution is the Schwarzschild metric with zero torsion.