A Concurrent Generalized Kropina Change
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917074848186368 |
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| author | Soleiman, A. Taha, Ebtsam H. |
| author_facet | Soleiman, A. Taha, Ebtsam H. |
| contents | This paper investigates a generalized Kropina metric featuring a specific $π$-form. Start with a Finsler manifold $(M,F)$ admits a concurrent $π$-vector field $\overlineφ$, then, examine the $ϕ$-concurrent generalized Kropina change defined by $\widehat{F}=\frac{F^{m+1}}{Φ^{m}},\,\, Φ^{m}>0$, where $Φ$ represents the corresponding $1$-form. We investigate the fundamental geometric objects associated with $\widehat{F}$ in an intrinsic manner after adopting this modification and present an example of a Finsler metric that admits a concurrent vector field along with $\widehat{F}$. Also, we prove that the geodesic sprays of $F$ and $\widehat{F}$ can never be projectively related. Moreover, we show $\overlineφ$ is not concurrent with respect to $\widehat{F}$. Eventhough, we give a sufficient condition for $\overlineφ$ to be concurrent with respect to $\widehat{F}$. Finally, we prove that the $ϕ$-concurrent generalized Kropina change ($F \longrightarrow \widehat{F}$) preserves the almost rational property of the initial Finsler metric ${F}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06000 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Concurrent Generalized Kropina Change Soleiman, A. Taha, Ebtsam H. Differential Geometry This paper investigates a generalized Kropina metric featuring a specific $π$-form. Start with a Finsler manifold $(M,F)$ admits a concurrent $π$-vector field $\overlineφ$, then, examine the $ϕ$-concurrent generalized Kropina change defined by $\widehat{F}=\frac{F^{m+1}}{Φ^{m}},\,\, Φ^{m}>0$, where $Φ$ represents the corresponding $1$-form. We investigate the fundamental geometric objects associated with $\widehat{F}$ in an intrinsic manner after adopting this modification and present an example of a Finsler metric that admits a concurrent vector field along with $\widehat{F}$. Also, we prove that the geodesic sprays of $F$ and $\widehat{F}$ can never be projectively related. Moreover, we show $\overlineφ$ is not concurrent with respect to $\widehat{F}$. Eventhough, we give a sufficient condition for $\overlineφ$ to be concurrent with respect to $\widehat{F}$. Finally, we prove that the $ϕ$-concurrent generalized Kropina change ($F \longrightarrow \widehat{F}$) preserves the almost rational property of the initial Finsler metric ${F}$. |
| title | A Concurrent Generalized Kropina Change |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.06000 |