The T-tensor of spherically symmetric Finsler metrics

TL;DR

This paper derives a general expression for the T-tensor of spherically symmetric Finsler metrics, characterizes those satisfying the T-condition, and explores their relation to quasi-C-reducibility based on the mean Cartan tensor.

Contribution

It provides a new explicit formula for the T-tensor in spherically symmetric Finsler metrics and characterizes metrics satisfying the T-condition, linking them to quasi-C-reducibility.

Findings

Derived a general expression for the T-tensor in terms of (r,s) and derivatives

Characterized all spherically symmetric Finsler metrics with vanishing T-tensor

Showed that metrics with non-zero mean Cartan tensor are quasi-C-reducible

Abstract

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 $n \geq 3$, with a non-zero mean Cartan tensor are quasi-C-reducible.