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

Abstract

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 ∇¯ 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,∇¯,G) is a statistical manifold and one family of proper general natural metrics such that (TM∖{0},∇¯,G) is a quasi-statistical manifold.