Curvatures of tangent bundle of Finsler manifold with Cheeger-Gromoll metric

Z. Raei, D. Latifi

Let $(M,F)$ be a Finsler manifold and $G$ be the Cheeger-Gromoll metric induced by $F$ on the slit tangent bundle $\widetilde{TM}=TM\backslash 0$. In this paper, we will prove that the Finsler manifold $(M,F)$ is of scalar flag curvature $K=\alpha$ if and only if the unit horizontal Liouville vector field $\xi=\frac{y^i}{F}\frac{\delta}{\delta x^i}$ is a Killing vector field on the indicatrix bundle $IM$ where $\alpha: TM\rightarrow R$ is defined by $\alpha(x,y)=1+g_x(y,y)$. Also, we will calculate the scalar curvature of a tangent bundle equipped with Cheeger-Gromoll metric and obtain some conditions for the scalar curvature to be a positively homogeneous function of degree zero with respect to the fiber coordinates of $\widetilde{TM}$.