Quasi-Statistical Schouten-van Kampen Connections on the Tangent Bundle

We determine the general natural metrics G on the total space TM of the tangent bundle of a Riemannian manifold (M, g) such that the Schouten-van Kampen connection (del) over bar associated to the Levi-Civita connection of G is (quasi-)statistical. We prove that the base manifold must be a space form and in particular, when G is a natural diagonal metric, (M, g) must be locally flat. We prove that there exist one family of natural diagonal metrics and two families of proper general natural metrics such that (TM, (del) over bar, G) is a statistical manifold and one family of proper general natural metrics such that (TM \ {0}, (del) over bar, G) is a quasi-statistical manifold.