On the geometry of the rescaled Riemannian metric on tensor bundles of arbitrary type

Abstract

Let $(M,g)$ be an $n-$dimensional Riemannian manifold and $T_{1}^{1}(M)$ be its $(1,1)-$tensor bundle equipped with the rescaled Sasaki type metric $% ^{S}g_{f}$ which rescale the horizontal part by a nonzero differentiable function $f$. In the present paper, we discuss curvature properties of the Levi-Civita connection and another metric connection of $T_{1}^{1}(M)$. We construct almost paracomplex Norden structures on $T_{1}^{1}(M)$ and investigate conditions for these structures to be para-Kähler (paraholomorphic) and quasi-Kähler. Also, some properties of almost paracomplex Norden structures in context of almost product Riemannian manifolds are presented. Finally we introduce the rescaled Sasaki type metric $^{S}g_{f}$ on the $(p,q)-$\ tensor bundle and characterize the geodesics on the $(p,q)$-tensor bundle with respect to the Levi-Civita connection of \textit{${}$}$^{S}g_{f}$ and another metric connection of \textit{${}$}$^{S}g_{f}.$