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

Abstract

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