$(\alpha ,\beta )$-metrics satisfying the $T$-Condition or the $\sigma T$-Condition

Abstract

We describe the $(\alpha ,\beta )$-metrics whose the $T$-tensor vanishes ($T$-condition) and the $(\alpha ,\beta )$-metrics that satisfy the $\sigma T$-condition ${\sigma }_h{T}_{ijk}^h=0$, where ${\sigma }_h=\frac{\partial \sigma }{\partial x^h}$ and $\sigma$ is a smooth function on $M$. These classes have already been obtained by Z. Shen and G. S. Asanov in a completely different approach. The Finsler metrics of the first class are Berwaldian, the metrics of the second class are almost regular non-Berwaldian Landsberg metrics.

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