A realization problem in statistical and affine geometry

A few realization results are proved in statistical and affine geometry in the 2-dimensional case. For instance, it is proved that each analytic metric tensor field on a 2-dimensional manifold M can be locally realized as the Blaschke metric on a Blaschke surface. It is not true, however, that each torsion-free connection (even analytic) can be locally realized on a Blaschke surface. But each analytic torsion-free, Ricci-symmetric, projectively flat connection on M can be locally realized as the dual connection on a Blaschke surface. Equiaffine versions of these theorems are also proved. A generalization of Amari-Armstrong theorem is proved. Namely, we prove that each analytic metric tensor field and an analytic 2-covariant tensor field whose anti-symmetric part is closed can be locally realized as a metric of a statistical structure and the Ricci tensor of the statistical connection of this structure, respectively.