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