Matsumoto Metrics of Reversible Curvature

TL;DR

This paper investigates the conditions under which Matsumoto metrics exhibit reversible Riemann and Ricci curvature, establishing equivalences with quadratic curvature conditions and linking weakly Einstein metrics to Ricci-reversibility.

Contribution

It provides new characterizations of reversibility in Matsumoto metrics, showing equivalences with quadratic curvature properties and connecting Einstein conditions to Ricci-reversibility.

Findings

Matsumoto metric is R-reversible iff R-quadratic

Matsumoto metric is Ricci-reversible iff Ricci-quadratic

Weakly Einstein Matsumoto metrics are Ricci-reversible

Abstract

In this paper, we study the reversibility of Riemann curvature and Ricci curvature for the Matsumoto metric and prove three global results. First, we prove that a Matsumoto metric is R-reversible if and only if it is R-quadratic. Then we show that a Matsumoto metric is Ricci-reversible if and only if it is Ricci-quadratic. Finally, we prove that every weakly Einstein Matsumoto metric is Ricci-reversible.