The T-tensor of spherically symmetric Finsler metrics
Salah G. Elgendi
TL;DR
This work derives an explicit T-tensor for spherically symmetric Finsler metrics $F=u\phi(r,s)$ on $\mathbb{R}^n$, expressing it in terms of $\phi$ and its derivatives via $\Phi$, $\Psi$, and $\Omega$ alongside geometric tensors $\hbar_{ij}$ and $m_i$. It also computes the Cartan and mean Cartan tensors, proving that if $n\ge3$ and the mean Cartan tensor is nonzero, the metric is quasi-C-reducible. A complete classification of the T-condition is given: either the metric is Riemannian ($\sigma_2=0$) or $\phi$ must take a specific separable form $\phi(r,s)=a(r)\, s^{\frac{c(r) r^2-1}{c(r) r^2}}( r^2 - s^2)^{\frac{1}{2 c(r) r^2}}$, with a converse established. The paper also provides explicit T-tensor expressions for Kropina-type and Randers-type examples and discusses computational approaches (e.g., Maple) for these symbolic results, contributing concrete invariants for symmetric Finsler spaces and informing geometric classifications relevant to generalized gravity models.
Abstract
This paper is devoted to the study of the T-tensor associated with a spherically symmetric Finsler metric $F=uφ(r,s)$ on \(\mathbb{R}^n\). We derive a general expression for the T-tensor in terms of the scalar function \(φ(r, s)\) 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.