Generalized P-Reducible $(\alpha, \beta)$-Metrics with Vanishing S-curvature

TL;DR

This paper investigates a class of Finsler metrics called generalized P-reducible metrics with vanishing S-curvature, proving they must be either Berwald or C-reducible, thus addressing an open problem in Finsler geometry.

Contribution

It introduces and analyzes generalized P-reducible $(eta, eta)$-metrics with vanishing S-curvature, showing they are either Berwald or C-reducible, resolving a longstanding open problem.

Findings

Every generalized P-reducible $(eta, eta)$-metric with vanishing S-curvature is either Berwald or C-reducible.

There are no non-trivial P-reducible $(eta, eta)$-metrics with vanishing S-curvature.

The study addresses an open problem posed by Matsumoto-Shimada.

Abstract

In this paper, we study one of the open problems in Finsler geometry which presented by Matsumoto-Shimada about the existence of P-reducible metric which is not C-reducible. For this aim, we study a class of Finsler metrics called generalized P-reducible metrics that contains the class of P-reducible metrics. We prove that every generalized P-reducible $(\alpha, \beta)$-metric with vanishing S-curvature reduces to a Berwald metric or C-reducible metric. It results that there is not any concrete P-reducible $(\alpha,\beta)$-metric with vanishing S-curvature.