Projective flatness of cylindrically symmetric Finsler metrics
Abstract
In this paper, we study cylindrically symmetric projectively flat Finsler metrics F=|y̅|ϕ(x0,r,s,z). We reformulate the conditions for projective flatness and strong convexity in terms of Ω=ϕ−sϕs−zϕz, isolating the compatibility condition missing in earlier work. Using the method of characteristics, we give an integral formula for all smooth local solutions near points with r>0 and s=z=0. These solutions are determined by a characteristic potential up to the addition of an invariant exact one-form. In the reversible case, the equation for the potential reduces to a (2+1)-dimensional wave equation on a half-space. We illustrate the reconstruction with a polynomial potential and compute the potentials of a known family of spherically symmetric metrics and their linear pullbacks. The convexity conditions also admit decreasing profiles and recover the spherical Beltrami metric as a particular case. Finally, suitable positive sums of linear pullbacks yield smooth reversible projectively flat Finsler metrics on ℝn+1 whose tangent norm at the origin is non-Riemannian. Consequently, these metrics are not linear pullbacks of spherically symmetric metrics
Related
- Spherically symmetric Finsler metrics satisfying the σT-condition
- Ideal Poisson--Voronoi tessellations for Finsler metrics on symmetric spaces
- Kähler-Finsler rigidity of the Kobayashi metric and the Kähler base of weakly Kähler-Finsler metrics
- A Demailly-type approximation of singular Finsler metrics
- Conformal boundary rigidity for simple Finsler metrics
BibTeX
@misc{solorzano2026projective,
title = {{Projective flatness of cylindrically symmetric Finsler metrics}},
author = {Newton Solórzano and Víctor León and Dik Lujerio},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08747},
eprint = {2611.08747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08747},
}
RIS
TY - UNPB
TI - Projective flatness of cylindrically symmetric Finsler metrics
AU - Newton Solórzano
AU - Víctor León
AU - Dik Lujerio
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08747
UR - https://arxiv.org/abs/2611.08747
AB - In this paper, we study cylindrically symmetric projectively flat Finsler metrics F=|y̅|ϕ(x^0,r,s,z). We reformulate the conditions for projective flatness and strong convexity in terms of Ω=ϕ−sϕ_s−zϕ_z, isolating the compatibility condition missing in earlier work. Using the method of characteristics, we give an integral formula for all smooth local solutions near points with r>0 and s=z=0. These solutions are determined by a characteristic potential up to the addition of an invariant exact one-form. In the reversible case, the equation for the potential reduces to a (2+1)-dimensional wave equation on a half-space. We illustrate the reconstruction with a polynomial potential and compute the potentials of a known family of spherically symmetric metrics and their linear pullbacks. The convexity conditions also admit decreasing profiles and recover the spherical Beltrami metric as a particular case. Finally, suitable positive sums of linear pullbacks yield smooth reversible projectively flat Finsler metrics on ℝ^n+1 whose tangent norm at the origin is non-Riemannian. Consequently, these metrics are not linear pullbacks of spherically symmetric metrics
ER -