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

Let (M,g) be an n-dimensional Riemannian manifold and T11(M) be its (1,1)-tensor bundle equipped with the rescaled Sasaki type metric Sgf which rescale the horizontal part by a non-zero differentiable function f. In the present paper, we discuss curvature properties of the Levi-Civita connection and another metric connection of T11(M). We construct almost product Riemannian structures on T11(M) and investigate conditions for these structures to be locally decomposable. Also, some applications concerning with these almost product Riemannian structures on T11(M) are presented. Finally we introduce the rescaled Sasaki type metric Sgf on the (p,q)-tensor bundle and characterize the geodesics on the (p,q)-tensor bundle with respect to the Levi-Civita connection and another metric connection of Sgf.