The curvatures of spherically symmetric Finsler metrics in $R^n$

TL;DR

This paper classifies spherically symmetric Finsler metrics in Euclidean space, proving non-existence of certain types, deriving PDE systems for special curvature conditions, and providing explicit examples of metrics with constant flag curvature.

Contribution

It offers a complete classification of spherically symmetric Berwald and Landsberg metrics, derives PDE systems for constant flag curvature and Einstein metrics, and constructs explicit non-Randers examples.

Findings

No non-Berwald Landsberg metrics among regular cases.

Derived PDE systems for constant flag curvature and Einstein metrics.

Constructed explicit non-projective, non-Randers Finsler metrics with constant flag curvature.

Abstract

In this paper, we classify the spherically symmetric Berwald metrics in $\mathbb{R}^n$. For the spherically symmetric Landsberg metrics, we prove that there do not exist any non-Berwald metrics among the regular case. The partial differential equation systems which can respectively characterize the spherically symmetric Finsler metrics with constant flag curvature and Einstein metrics of this type is also obtained. Utilizing these equations, we find an effective way to construct the non-projective, non-Randers Finsler metrics with constant flag curvature and many explicit examples are given by this method.