@article{Elgendi2024, 
author = {Salah G. Elgendi and Zoltán Muzsnay},
title = {The geometry of geodesic invariant functions and applications to Landsberg surfaces},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {9},
pages = {23617-23631},
keywords = {spray, holonomy distribution, S-invariant functions (first integrals), Landsberg surfaces, flag curvature},
url = {https://www.sciopen.com/article/10.3934/math.20241148},
doi = {10.3934/math.20241148},
abstract = {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  P, we associated a vertical subdistribution  VP and found the relation between the holonomy distribution and  VP by showing that the vertical part of the holonomy distribution is the intersection of all spaces  VFS associated with  FS where  FS 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.}
}