Musical Isomorphisms and Statistical Manifolds

Let ( M , g , ∇ ( α ) ) be a statistical manifold and g ♭ : T M → T ∗ M be a musical isomorphism from the tangent bundle onto the cotangent bundle. Using the α -vertical and α -horizontal lifts on the tangent bundle of the statistical manifold M , we construct the g - α -vertical and g - α -horizontal lifts on the cotangent bundle with the aid of the musical isomorphism g ♭ . We prove that the Lie bracket of the α -horizontal lifts of vector fields to tangent and cotangent bundles is g ♭ -related if and only if the α -curvature tensor is an even function of α . Also, we get statistical structures via the musical isomorphism in the cotangent bundle. Finally, we give the notion of the Schouten–Van Kampen ( α ) -connection associated with the statistical connection on the cotangent bundles. Furthermore, we provide some non-trivial examples as applications to this study.