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

TL;DR

This paper investigates metric connections with torsion on the cotangent bundle of a manifold, characterizing special vector fields and curvature conditions under a modified Riemannian extension.

Contribution

It introduces a new study of metric connections with torsion on cotangent bundles equipped with a modified Riemannian extension, including characterizations and curvature conditions.

Findings

Characterization of fibre-preserving projective vector fields.

Conditions for semi-symmetry and related properties.

Results on the Schouten-Van Kampen connection.

Abstract

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 $(T^{\ast }M,{}\bar{g}%_{\nabla ,c})$ 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 $% \widetilde{\nabla }$. Finally, we present some results concerning the Schouten-Van Kampen connection associated to the Levi-Civita connection $% \bar{\nabla }$ of the modified Riemannian extension $\bar{g}%_{\nabla ,c}$.