Vanishing of Killing vector fields on compact Finsler manifolds (Q725630)

Let $(M,F)$ be an $n$-dimensional Finsler manifold. In the paper under review the author introduces the notion of mean Ricci curvature $\widetilde{\mathrm{RIC}}$ on $(M,F)$ which is a kind of integral average of the flag curvature on the indicatrix $S_xM$ at each point. By definition \[ \widetilde{\mathrm{RIC}}(v):= \frac{1}{c_{n-1}}\int_{S_xM}F^{-2}R_{ij}v^iv^j\frac{\sqrt{\det g_{ij}}}{\sqrt{\det a_{ij}}} dv, \] where $a_{ij}$ are the components of the related Riemannian metric defined in [\textit{V. S. Matveev} et al., Ann. Inst. Fourier 59, No. 3, 937--949 (2009; Zbl 1179.53075)]. Here, the author develops Riemannian techniques and obtains several results. Meanwhile it is shown that in a compact Finsler manifold with negative flag curvature, there is no non-trivial Killing field. The goal of this paper is to show that on every compact Finsler manifold with non-positive mean Ricci curvature $\widetilde{\mathrm{RIC}}\leq0$, the Killing fields $V$ are parallel, and $\widetilde{\mathrm{RIC}}(V,V)=0$. Furthermore, if the mean Ricci curvature is negative, then there is no non-trivial Killing field.