Statistical manifolds with almost contact structures and its statistical submersions (Q860239)

In this paper the author considers statistical manifolds with almost contact structure which are Sasaki-like, and Sasaki-like statistical submersions. As a statistical manifold, \(M\) is a semi-Riemannian manifold with metric \(g\) and a torsion-free affine connection \(\nabla \) such that \(\nabla g\) is symmetric. Also defined on \(( M,g,\nabla) \) is a conjugate affine connection \(\nabla ^{\ast }\) such that \(\nabla ^{\ast }g\) is symmetric and \(( \nabla ^{\ast }) ^{\ast }=\nabla \). The author obtains the result that if \(\pi :( M,\nabla ,g) \rightarrow ( B,\widehat{\nabla },g_{B}) \) is a Sasaki-like statistical submersion then the base space is a Kähler-like statistical manifold and each fiber is a Sasaki-like statistical manifold. For the main results in the paper the curvature tensor with respect to \(\nabla \) of the total space is assumed to satisfy a certain condition (note that the same condition on a Sasakian manifold would imply constant holomorphic sectional curvature \(c\)). The author then obtains the result that, if \(\pi :( M,\nabla ,g) \rightarrow ( B,\widehat{\nabla },g_{B}) \) is a Sasaki-like statistical submersion, then the curvature tensor of the base space satisfies the same condition as on a Kähler manifold with constant holomorphic sectional curvature \(c+3.\) The author also obtains related results with additional conditions on the mean curvature vector of the fiber with respect to \(\nabla \) and to \(\nabla ^{\ast }.\)