@article{Elgendi2024, 
author = {Salah G. Elgendi},
title = {Parallel one forms on special Finsler manifolds},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {12},
pages = {34356-34371},
keywords = {parallel 1-form, Landsberg manifolds, scalar curvature, general Berwald's metric, spherically symmetric metrics},
url = {https://www.sciopen.com/article/10.3934/math.20241636},
doi = {10.3934/math.20241636},
abstract = {In this paper, we investigated the existence of parallel 1-forms on specific Finsler manifolds. We demonstrated that Landsberg manifolds admitting a parallel 1-form had a mean Berwald curvature of rank of at most  n−2. As a result, Landsberg surfaces with parallel 1-forms were necessarily Berwaldian. We further established that the metrizability freedom of the geodesic spray for Landsberg metrics with parallel 1-forms was at least  2. We figured out that some special Finsler metrics did not admit a parallel 1-form. Specifically, no parallel 1-form was admitted for any Finsler metrics of nonvanishing scalar curvature, among them the projectively flat metrics with nonvanishing scalar curvature. Furthermore, neither the general Berwald's metric nor the non-Riemannian spherically symmetric metrics admited a parallel 1-form. Consequently, we observed that certain  (α,β)-metrics and generalized  (α,β)-metrics did not admit parallel 1-forms.}
}