Golden Riemannian structures on the tangent bundle with g-natural metrics

Abstract

Starting from the-natural Riemannian metric G on the tangent bundle TM of a Riemannian manifold (M,), we construct a family of the Golden Riemannian structures ϕ on the tangent bundle (TM, G). Then we investigate the integrability of such Golden Riemannian structures on the tangent bundle TM and show that there is a direct correlation between the locally decomposable property of (TM, ϕ, G) and the locally flatness of manifold (M,).