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

TL;DR

This paper explores the geometric properties of the rescaled Sasaki type metric on tensor bundles over Riemannian manifolds, including curvature, complex structures, and geodesics, extending understanding of tensor bundle geometry.

Contribution

It introduces and analyzes the curvature, complex structures, and geodesic properties of the rescaled Sasaki type metric on tensor bundles of arbitrary type.

Findings

Curvature properties of the Levi-Civita and other metric connections are characterized.

Conditions for almost paracomplex Norden structures to be para-Kähler or quasi-Kähler are established.

Geodesic equations are derived for the tensor bundles with respect to the rescaled metric.

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-K\"{a}hler (paraholomorphic) and quasi-K\"{a}hler. 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{${}$}$^{S}g_{f}.$