A note on "On the classification of Landsberg spherically symmetric Finsler metrics"

TL;DR

This paper proves that all spherically symmetric Landsberg surfaces are Berwaldian and classifies Berwald spherically symmetric Finsler metrics of dimension n≥3, showing they are either Riemannian or follow a specific formula.

Contribution

It establishes that all spherically symmetric Landsberg surfaces are Berwaldian and classifies Berwald spherically symmetric Finsler metrics in higher dimensions.

Findings

All Landsberg spherically symmetric surfaces are Berwaldian.

Berwald spherically symmetric metrics of dimension n≥3 are either Riemannian or follow a specific formula.

Spherically symmetric metrics with a homogeneous degree -1 function in r and s are Berwaldian.

Abstract

In this paper, we prove that all spherically symmetric Landsberg surfaces are Berwaldian. We modify the classification of spherically symmetric Finsler metrics, done by the author in [S. G. Elgendi, On the classification of Landsberg spherically symmetric Finsler metrics, Int. J. Geom. Methods Mod. Phys. 18 (2021)], of Berwald type of dimension $n\geq 3$. Precisely, we show that all Berwald spherically symmetric metrics of dimension $n\geq 3$ are Riemannian or given by a certain formula. As a simple class of Berwaldian metrics, we prove that all spherically symmetric metrics in which the function $\phi$ is homogeneous of degree $-1$ in $r$ and $s$ are Berwaldian.