Geometry of dual spaces of affine-metric connection

In this work, we consider the dual geometry of a normalized space of affine connection $\mathrm A_{n,n}$. In particular, we study the dual spaces of an affine-metric connection $\overset{p}{\mathrm M}_{n,n}$, which are induced by a nondegenerate normalization of a space of affine-metric connection $\mathrm M_{n,n}$.