Parallel one forms on special Finsler manifolds

TL;DR

This paper explores the conditions under which parallel 1-forms exist on special Finsler manifolds, revealing restrictions on their curvature and metric types, and establishing that certain classes do not admit such forms.

Contribution

It provides new results on the existence and non-existence of parallel 1-forms on various Finsler manifolds, including Landsberg, Berwald, and $(eta)$-metrics.

Findings

Landsberg manifolds with parallel 1-forms have mean Berwald curvature rank ≤ n-2

Landsberg surfaces with parallel 1-forms are necessarily Berwaldian

Certain Finsler metrics, including non-vanishing scalar curvature and some $(eta)$-metrics, do not admit parallel 1-forms

Abstract

In this paper, we investigate the existence of parallel 1-forms on specific Finsler manifolds. We demonstrate that Landsberg manifolds admitting a parallel 1-form have a mean Berwald curvature of rank at most $n-2$. As a result, Landsberg surfaces with parallel 1-forms are necessarily Berwaldian. We further establish that the metrizability freedom of the geodesic spray for Landsberg metrics with parallel 1-forms is at least $2$. We figure out that some special Finsler metrics do not admit a parallel 1-form. Specifically, no parallel 1-form is admitted for any Finsler metrics of non-vanishing scalar curvature, among them the projectively flat metrics with non-vanishing scalar curvature. Furthermore, neither the general Berwald's metric nor the non-Riemannian spherically symmetric metrics admit a parallel 1-form. Consequently, we observe that certain $(\alpha,\beta)$-metrics and generalized $(\alpha,\beta)$-metrics do not admit parallel 1-forms.