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

TL;DR

This paper classifies certain $(eta)$-metrics in Finsler geometry that satisfy specific tensor vanishing conditions, identifying their geometric properties and linking to previous classifications by Shen and Asanov.

Contribution

It provides a new characterization of $(eta)$-metrics satisfying the $T$-condition and $\sigma T$-condition, connecting these classes to known geometric structures.

Findings

Metrics satisfying the $T$-condition are Berwaldian.

Metrics satisfying the $\sigma T$-condition are almost regular non-Berwaldian Landsberg metrics.

The classes were previously obtained by Shen and Asanov through different methods.

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 $\sigma_hT^h_{ijk}=0$, 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.