Related papers: Hilbert forms for a Finsler metrizable projective …
This paper is concerned with the problem of determining whether a projective-equivalence class of sprays is the geodesic class of a Finsler function. We address both the local and the global aspects of this problem. We present our results…
The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective…
Geodesics, which play an important role in spray-Finsler geometry, are integral curves of a spray vector field on a manifold. Some comparison theorems and rigidity issues are established on the completeness of geodesics of a spray or a…
In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used…
This paper studies spherically symmetric sprays, i.e., sprays that are invariant under orthogonal transformations. We first establish a canonical form for such sprays, showing that their geodesic coefficients can be expressed as \(G^i =…
In the previous work, the notion of the Finsleroid--Finsler space have been formulated and the necessary and sufficient conditions for the space to be of the Landsberg type have been found. In the present paper, starting with particular…
In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively…
Locally projectively flat metrics (or sprays) form a rich class of metrics (or sprays) in Finsler and spray geometry. The characterization of such metrics is the Hilbert Fourth Problem in the regular case. In this paper we study the…
It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In Theorem 3.1 we show that the isotropy…
In 2001, Zhongmin Shen asked if it is possible for two projectively related Finsler metrics to have the same Riemann curvature tensor, [14, page 184]. In this paper, we provide an answer to this question, within the class of Finsler metrics…
We prove that various Finsler metrizability problems for sprays can be reformulated in terms of the geodesic invariance of two tensors (metric and angular). We show that gyroscopic sprays is the the largest class of sprays with geodesic…
In this paper, we introduce the notion of one form deformation of sprays. The metrizability of the new spray, when the background spray is flat, is characterized. Therefore, we obtain new projectively flat metrics of constant flag curvature…
In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 \lambda F\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $\lambda \in…
We consider the projective Finsler metrizability problem: under what conditions the solutions of a given system of second-order ordinary differential equations (SODE) coincide with the geodesics of a Finsler metric, as oriented curves.…
In this paper we investigate the following question: under what conditions can a second-order homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the Euler-Lagrange partial…
In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$ and a holonomy invariant function $P$, we investigate the metrizability property of…
Every Finsler metric naturally induces a spray but not so for the converse. The notion for sprays of scalar (resp. isotropic) curvature has been known as a generalization for Finsler metrics of scalar (resp. isotropic) flag curvature. In…
For a Finsler metric $F$, we introduce the notion of $F$-covariant coefficients $H_i$ of the geodesic spray of $F$ (Def. 3.1). We study some geometric consequences concerning the objects $H_i$. If the $F$-covariant coefficients $H_i$ are…
In the asymmetric setting, Hilbert's fourth problem asks to construct and study all (non-reversible) projective Finsler metrics: Finsler metrics defined on open, convex subsets of real projective $n$-space for which geodesics lie on…
This paper explores the generalized projective Riemann curvature in Finsler geometry, focusing on the properties of projectively equivalent Finsler metrics and the invariance of their curvature structures under projective transformations.…