Almost product structures on statistical manifolds and para-Kähler-like statistical submersions (Q2046106)

This paper is devoted to the study of statistical manifolds endowed with almost product structures. Recall that a statistical manifold is a semi-Riemannian manifold \((M,g)\) equipped with an additional structure given by a pair of torsion-free affine connections \((\nabla,\nabla^*)\) which are dual with respect to \(g\), i.e., such that \(X g(Y,Z) = g(\nabla_X Y,Z)+g(Y,\nabla^*_X Z)\) for all smooth tangent vector fields on \(M\). The pair \((\nabla,g)\) is called a statistical structure on \(M\) if \(\nabla g\) is symmetric. The statistical structure \((\nabla,g)\) is said to be a Hessian structure if \(\nabla\) is flat. The concept of para-Kähler-like statistical manifold somehow generalizes the notion of para-Kähler manifold, as explained in Section 3. The author proves that the statistical structure of a para-Kähler-like statistical manifold of constant curvature in the sense of Kurose (i.e., such that \({}^\nabla\! R(X,Y)Z = k\left\{g(Y,Z)X-g(X,Z)Y\right\}\) for all smooth vector fields on \(M\)) is a Hessian structure. He also derives the main properties of statistical submersions which are compatible with almost product structures. Several illustrative examples are given in Section 5. Apart from examples constructed explicitly in coordinates, he uses the Sasaki metric to provide para-Kähler-like statistical structures on the tangent bundle of para-Hermitian-like manifolds, and shows that the statistical manifolds corresponding to some standard statistical models can be equipped with such structures.