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

Abstract

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