Parallel one forms on special Finsler manifolds

Salah G. Elgendi; Salah G. Elgendi

doi:10.3934/math.20241636

AIMS Mathematics

2024, Volume 9, Issue 12: 34356-34371. doi: 10.3934/math.20241636

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Research article

Parallel one forms on special Finsler manifolds

Salah G. Elgendi ^,

Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, Saudi Arabia

Received: 04 October 2024 Revised: 25 November 2024 Accepted: 29 November 2024 Published: 05 December 2024
MSC : 53B40, 53C60

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.
- parallel 1-form,
- Landsberg manifolds,
- scalar curvature,
- general Berwald's metric,
- spherically symmetric metrics
Citation: Salah G. Elgendi. Parallel one forms on special Finsler manifolds[J]. AIMS Mathematics, 2024, 9(12): 34356-34371. doi: 10.3934/math.20241636

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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 $ (\alpha, \beta) $-metrics and generalized $ (\alpha, \beta) $-metrics did not admit parallel 1-forms.

References

[1]	P. Percell, Parallel vector fields on manifolds with boundary, J. Differ. Geom., 16 (1981), 101–104. https://doi.org/10.4310/jdg/1214435992 doi: 10.4310/jdg/1214435992
[2]	C. Shibata, On invariant tensors of $\beta$-changes of Finsler metrics, J. Math. Kyoto U., 24 (1984), 163–188. https://doi.org/10.1215/kjm/1250521391 doi: 10.1215/kjm/1250521391
[3]	I. Mahara, Parallel vector fields and Einstein equations of gravity, Rawanda J., 20 (2011), 106–114.
[4]	L. Kozma, S. G. Elgendi, On the existence of parallel one forms, Int. J. Geom. Methods M., 20 (2023), 2350118. https://doi.org/10.1142/S0219887823501189 doi: 10.1142/S0219887823501189
[5]	Z. Shen, Projectively flat Finsler metrics of constant flag curvature, T. Am. Math. Soc., 255 (2003), 1713–1728. https://doi.org/10.1090/s0002-9947-02-03216-6 doi: 10.1090/s0002-9947-02-03216-6
[6]	L. Berwald, $\ddot{U}$berdien-dimensionalen Geometrien konstanter Kr$\ddot{u}$mmung, in denen die Geraden die K$\ddot{u}$rzesten sind, Math. Z., 30 (1929), 449–469. https://doi.org/10.1007/bf01187782 doi: 10.1007/bf01187782
[7]	J. Grifone, Structure presque-tangente et connexions. Ⅰ, Ann. I. Fourier, 22 (1972), 287–334. https://doi.org/10.5802/aif.407 doi: 10.5802/aif.407
[8]	S. G. Elgendi, Z. Muzsnay, The geometry of geodesic invariant functions and applications to Landsberg surfaces, AIMS Math., 9 (2024), 23617–23631. https://doi.org/10.3934/math.20241148 doi: 10.3934/math.20241148
[9]	Z. Shen, Differential geometry of spray and Finsler spaces, Springer, 2001. https://doi.org/10.1007/978-94-015-9727-2
[10]	S. S. Chern, Z. Shen, Riemann-Finsler geometry, World Scientific Publishers, 2004. https://doi.org/10.1142/5263
[11]	Z. Muzsnay, The Euler-Lagrange PDE and Finsler metrizability, Houston J. Math., 32 (2006), 79–98.
[12]	M. Crampin, A condition for a Landsberg space to be Berwaldian, Publ. Math.-Debrecen, 93 (2018), 143–155. https://doi.org/10.5486/pmd.2018.8111 doi: 10.5486/pmd.2018.8111
[13]	S. G. Elgendi, Z. Muzsnay, Freedom of h(2)-variationality and metrizability of sprays, Differ. Geom. Appl., 54 (2017), 194–207. https://doi.org/10.1016/j.difgeo.2017.03.020 doi: 10.1016/j.difgeo.2017.03.020
[14]	S. G. Elgendi, On the classification of Landsberg spherically symmetric Finsler metrics, Int. J. Geom. Methods M., 18 (2021), 2150232. https://doi.org/10.1142/s0219887821502327 doi: 10.1142/s0219887821502327
[15]	S. G. Elgendi, On the classification of Landsberg spherically symmetric Finsler metrics, Int. J. Geom. Methods M., 20 (2023), 2350096. https://doi.org/10.1142/s0219887823500962 doi: 10.1142/s0219887823500962
[16]	E. Guo, X. Mo, The geometry of spherically symmetric Finsler manifolds, Springer, 2018. https://doi.org/10.1007/978-981-13-1598-5
[17]	N. L. Youssef, S. H. Abed, S. G. Elgendi, Generalized $\beta$-conformal change of Finsler metrics, Int. J. Geom. Methods M, 7 (2010), 565–582. https://doi.org/10.1142/S021988781000444 doi: 10.1142/S021988781000444

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