On Riemann curvature of spherically symmetric metrics

TL;DR

This paper investigates the Riemann curvature properties of spherically symmetric Finsler metrics, establishing compatibility conditions, characterizing scalar curvature cases, and constructing geometric frames with examples.

Contribution

It introduces a curvature compatibility condition for spherically symmetric Finsler metrics and characterizes those with scalar curvature, advancing the understanding of their geometric structure.

Findings

Derived a curvature compatibility condition for spherically symmetric Finsler metrics

Characterized spherically symmetric metrics with scalar curvature

Constructed a Berwald frame and provided multiple examples

Abstract

In this paper, studying the inverse problem, we establish a curvature compatibility condition on a spherically symmetric Finsler metric. As an application, we characterize the spherically symmetric metrics of scalar curvature. We construct a Berwald frame for a spherically symmetric Finsler surface and calculate some associated geometric objects. Several examples are provided and discussed. Finally, we give a note on a certain general $(\alpha,\beta)$-metric that appears in the literature.