On the cotangent bundle with vertical modified riemannian extensions

Let M be an n-dimensional differentiable manifold with a torsion-free linear connection which induces on its cotangent bundle T^*M. The main purpose of the present paper is to study some properties of the vertical modified Riemannian extension on T^*M which is given as a new metric in 17. At first, we investigate a metric connection with torsion on T^*M. And then, we present the holomorphy properties with respect to a compatible almost complex structure. urthermore, we study locally decomposable Golden pseudo-Riemannian structures on the cotangent bundle endowed with vertical modified Riemannian extension.