Are torsion theories of gravitation equivalent to metric theories?

An investigation is made of gravitation theories which are based on a Lagrangian constructed from the scalar curvature, in which torsion is allowed and in which the metric compatibility with the connection is not assumed in general. It is shown that because of the invariance of the scalar curvature under a projective transformation of the connection, only that part of the connection that is projectively invariant is determined by the theory. The theory is then reformulated so that the total action functional (gravitation plus source fields) depends only on the projective invariant part of the connection. The resulting theory is shown to be metrizable and equivalent to Einstein's theory with a modified source Lagrangian.