In this paper, for a given spray $ S $ on an $ n $-dimensional manifold $ M $, we investigated the geometry of $ S $-invariant functions. For an $ S $-invariant function $ {\mathcal P} $, we associated a vertical subdistribution $ {{\mathcal V}}_{\mathcal P} $ and found the relation between the holonomy distribution and $ {{\mathcal V}}_{\mathcal P} $ by showing that the vertical part of the holonomy distribution is the intersection of all spaces $ {{\mathcal V}}_{ {\mathcal F}_S} $ associated with $ {\mathcal F}_S $ where $ {\mathcal F}_S $ is the set of all Finsler functions that have the geodesic spray $ S $. As an application, we studied the Landsberg Finsler surfaces. We proved that a Landsberg surface with $ S $-invariant flag curvature is Riemannian or has a vanishing flag curvature. We showed that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is $ S $-invariant if and only if it is constant; in this case, the surface is Riemannian. Finally, for a Berwald surface, we proved that the flag curvature is $ H $-invariant if and only if it is constant.
Citation: Salah G. Elgendi, Zoltán Muzsnay. The geometry of geodesic invariant functions and applications to Landsberg surfaces[J]. AIMS Mathematics, 2024, 9(9): 23617-23631. doi: 10.3934/math.20241148
In this paper, for a given spray $ S $ on an $ n $-dimensional manifold $ M $, we investigated the geometry of $ S $-invariant functions. For an $ S $-invariant function $ {\mathcal P} $, we associated a vertical subdistribution $ {{\mathcal V}}_{\mathcal P} $ and found the relation between the holonomy distribution and $ {{\mathcal V}}_{\mathcal P} $ by showing that the vertical part of the holonomy distribution is the intersection of all spaces $ {{\mathcal V}}_{ {\mathcal F}_S} $ associated with $ {\mathcal F}_S $ where $ {\mathcal F}_S $ is the set of all Finsler functions that have the geodesic spray $ S $. As an application, we studied the Landsberg Finsler surfaces. We proved that a Landsberg surface with $ S $-invariant flag curvature is Riemannian or has a vanishing flag curvature. We showed that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is $ S $-invariant if and only if it is constant; in this case, the surface is Riemannian. Finally, for a Berwald surface, we proved that the flag curvature is $ H $-invariant if and only if it is constant.
| [1] | D. Bao, On two curvature-driven problems in Riemann-Finsler geometry, Adv. Stud. Pure Math., 48 (2007), 19–71. |
| [2] |
I. Bucataru, G. Cretu, E. H. Taha, Frobenius integrability and Finsler metrizability for 2-dimensional sprays, Differ. Geom. Appl., 56 (2018), 308–324. https://doi.org/10.1016/j.difgeo.2017.10.002 doi: 10.1016/j.difgeo.2017.10.002
|
| [3] |
I. Bucataru, O. Constantinescu, G. Cretu, A class of Finsler metrics admitting first integrals, J. Geom. Phys., 166 (2021), 104254. https://doi.org/10.1016/j.geomphys.2021.104254 doi: 10.1016/j.geomphys.2021.104254
|
| [4] | I. Bucataru, Z. Muzsnay, Projective and Finsler metrizability: Parameterization-rigidity of the geodesics, Int. J. Math., 23 (2012). https://doi.org/10.1142/S0129167X12500991 |
| [5] |
S. Bacsó, M. Matsumoto, Reduction theorems of certain Landsberg spaces to Berwald spaces, Publ. Math.-Debrecen, 48 (1996), 357–366. https://doi.org/10.5486/pmd.1996.1733 doi: 10.5486/pmd.1996.1733
|
| [6] |
L. Berwald, On Finsler and Cartan geometries. Ⅲ: Two-dimensional Finsler spaces with rectilinear extremals, Ann. Math., 42 (1941), 84–112. https://doi.org/10.2307/1968989 doi: 10.2307/1968989
|
| [7] |
S. G. Elgendi, Solutions for the Landsberg unicorn problem in Finsler geometry, J. Geom. Phys., 159 (2021), 103918. https://doi.org/10.1016/j.geomphys.2020.103918 doi: 10.1016/j.geomphys.2020.103918
|
| [8] |
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
|
| [9] |
S. G. Elgendi, Z. Muzsnay, Metrizability of holonomy invariant projective deformation of sprays, Can. Math. Bull., 66 (2020), 701–714. https://doi.org/10.4153/s0008439520000016 doi: 10.4153/s0008439520000016
|
| [10] |
S. G. Elgendi, N. L. Youssef, A note on a result of L. Zhou's on Landsberg surfaces with $K = 0$ and $J = 0$, Differ. Geom. Appl., 77 (2021), 101779. https://doi.org/10.1016/j.difgeo.2021.101779 doi: 10.1016/j.difgeo.2021.101779
|
| [11] |
J. Grifone, Structure presque-tangente et connexions Ⅰ, Ann. Inst. Fourier, 22 (1972), 287–334. https://doi.org/10.5802/aif.407 doi: 10.5802/aif.407
|
| [12] | J. Grifone, Z. Muzsnay, Variational principles for second-order differential equations, World Scientific Publishing, 2000. https://doi.org/10.1142/9789812813596 |
| [13] | F. Ikeda, On two-dimensional Landsberg spaces, Tensor, 33 (1979), 43–48. |
| [14] |
P. Foulon, R. Ruggiero, A first integral for $C^{\infty}, k$-basic Finsler surfaces and application to rigidity, Proc. Am. Math. Soc., 144 (2016), 3847–3858. https://doi.org/10.1090/proc/13079 doi: 10.1090/proc/13079
|
| [15] | Z. Muzsnay, The Euler-Lagrange PDE and Finsler metrizability, Houston J. Math., 32 (2006), 79–98. |
| [16] | S. V. Sabau, H. Shimada, Riemann-Finsler surfaces, Math. Soc. Japan, 48 (2007), 125–163. https://doi.org/10.2969/aspm/04810125 |
| [17] |
Z. Shen, On a class of Landsberg metrics in Finsler geometry, Canad. J. Math., 61 (2009), 1357–1374. https://doi.org/10.4153/cjm-2009-064-9 doi: 10.4153/cjm-2009-064-9
|
| [18] |
E. H. Taha, On a class of Landsberg metrics in Finsler geometry, Int. J. Geom. Meth. Mod. Phys., 20 (2023), 2350002. https://doi.org/10.1142/s0219887823500020 doi: 10.1142/s0219887823500020
|
| [19] |
L. Zhou, The Finsler surface with $K = 0$ and $J = 0$, Differ. Geom. Appl., 35 (2014), 370–380. https://doi.org/10.1016/j.difgeo.2014.02.003 doi: 10.1016/j.difgeo.2014.02.003
|
Salah G. Elgendi, Zoltán Muzsnay. The geometry of geodesic invariant functions and applications to Landsberg surfaces[J]. AIMS Mathematics, 2024, 9(9): 23617-23631. doi: 10.3934/math.20241148
DownLoad: