Parallel one forms on special Finsler manifolds
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 Berwald...
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AIMS Press
2024-12-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.20241636 |
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| author | Salah G. Elgendi |
| author_facet | Salah G. Elgendi |
| author_sort | Salah G. Elgendi |
| collection | DOAJ |
| description | 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 $ (\alpha, \beta) $-metrics and generalized $ (\alpha, \beta) $-metrics did not admit parallel 1-forms. |
| format | Article |
| id | doaj-art-a3640be9c6a443b387ee0f27e73bfc78 |
| institution | Kabale University |
| issn | 2473-6988 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj-art-a3640be9c6a443b387ee0f27e73bfc782025-01-23T07:53:25ZengAIMS PressAIMS Mathematics2473-69882024-12-01912343563437110.3934/math.20241636Parallel one forms on special Finsler manifoldsSalah G. Elgendi0Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, Saudi ArabiaIn 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 $ (\alpha, \beta) $-metrics and generalized $ (\alpha, \beta) $-metrics did not admit parallel 1-forms.https://www.aimspress.com/article/doi/10.3934/math.20241636parallel 1-formlandsberg manifoldsscalar curvaturegeneral berwald's metricspherically symmetric metrics |
| spellingShingle | Salah G. Elgendi Parallel one forms on special Finsler manifolds AIMS Mathematics parallel 1-form landsberg manifolds scalar curvature general berwald's metric spherically symmetric metrics |
| title | Parallel one forms on special Finsler manifolds |
| title_full | Parallel one forms on special Finsler manifolds |
| title_fullStr | Parallel one forms on special Finsler manifolds |
| title_full_unstemmed | Parallel one forms on special Finsler manifolds |
| title_short | Parallel one forms on special Finsler manifolds |
| title_sort | parallel one forms on special finsler manifolds |
| topic | parallel 1-form landsberg manifolds scalar curvature general berwald's metric spherically symmetric metrics |
| url | https://www.aimspress.com/article/doi/10.3934/math.20241636 |
| work_keys_str_mv | AT salahgelgendi paralleloneformsonspecialfinslermanifolds |