The T-tensor of spherically symmetric Finsler metrics

This paper is devoted to the study of the T-tensor associated with a spherically symmetric Finsler metric $F=u\phi (r,s)$ on ${\mathbb{R}}^n$. We derive a general expression for the T-tensor in terms of the scalar function $\phi (r,s)$ and its partial derivatives. Furthermore, we characterize all spherically symmetric Finsler metrics satisfying the so-called T-condition, that is, those for which the T-tensor vanishes. In addition, we obtain the formula for the mean Cartan tensor and demonstrate that all spherically symmetric Finsler metrics of dimension $n\ge 3$, with a non-zero mean Cartan tensor are quasi-C-reducible.