DERIVATIVES WITH RESPECT TO HORIZONTAL AND VERTICAL LIFTS OF THE CHEEGER-GROMOLL METRIC $^{CG}g$ ON THE $(1,1)-$TENSOR BUNDLE $T_{1}^{1}(M)$

Abstract

In this paper, we define the Cheeger-Gromoll metric in the $(1,1)$ $-$tensor bundle $T_{1}^{1}(M)$, which is completely determined by its action on vector fields of type $X^{H}$ and $\omega ^{V}$. Later, we obtain the covarient and Lie derivatives applied to the Cheeger-Gromoll metric with respect to the horizontal and vertical lifts of vector and kovector fields, respectively.