On metric connections with torsion on the cotangent bundle with modified Riemannian extension

Let $M$ be an $n-$dimensional differentiable manifold equipped with a torsion-free linear connection $\nabla$ and $T^{\ast }M$ its cotangent bundle. The present paper aims to study a metric connection \widetilde{% \nabla } with nonvanishing torsion on $T^{\ast }M$ with modified Riemannian extension ${}\bar{g}_{\nabla ,c}$. First, we give a characterization of fibre-preserving projective vector fields on $(T^{\ast }M,{}\bar{g}$ with respect to the metric connection $\widetilde{\nabla }$. Secondly, we study conditions for $(T^{\ast }M,{}\bar{g}_{\nabla ,c})$ to be semi-symmetric, Ricci semi-symmetric, $\widetilde{Z}$ semi-symmetric or locally conharmonically flat with respect to the metric connection $$. Finally, we present some results concerning the Schouten-Van Kampen connection associated to the Levi-Civita connection $$ of the modified Riemannian extension $\bar{g}$.