Induced almost para-Kähler Einstein metrics on cotangent bundles

In earlier work, we have shown that for certain geometric structures on a smooth manifold M of dimension n, one obtains an almost para-Kähler–Einstein metric on a manifold A of dimension 2n associated to the structure on M. The geometry also associates a diffeomorphism between A and $T^*M$ to any torsion-free connection compatible with the geometric structure. Hence we can use this construction to associate to each compatible connection an almost para-Kähler–Einstein metric on $T^*M$. In this short article, we discuss the relation of these metrics to Patterson–Walker metrics and derive explicit formulae for them in the cases of projective, conformal and Grassmannian structures.