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 T₁¹(M) be its (1,1)-tensor bundle equipped with the rescaled Sasaki type metric ^Sg_f 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 ^Sg_f 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 ^Sg_f.