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 ∇ and T∗M its cotangent bundle. The present paper aims to study a metric connection ∇ ~ with nonvanishing torsion on T∗M with modified Riemannian extension g¯ ∇ , c. First, we give a characterization of fibre-preserving projective vector fields on (T∗M, g¯ ∇ , c) with respect to the metric connection ∇ ~. Secondly, we study conditions for (T∗M, g¯ ∇ , c) to be semi-symmetric, Ricci semi-symmetric, 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 g¯ ∇ , c. © 2018, Springer International Publishing AG, part of Springer Nature.