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

TLDR: In this article, a metric connection with nonvanishing torsion with modified Riemannian extension was studied and a characterization of fiber-preserving projective vector fields was given.

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 $$\overline{g} _{\nabla ,c}$$ .