arXiv research

Study on Berwald-Weyl curvature with projective invariance and vanishing results.

For a $2$-dimensional non-flat spray we associate a Berwald frame and a $3$-dimensional distribution that we call the Berwald distribution. The Frobenius integrability of the Berwald distribution characterises the Finsler metrizability of the given spray. In the integrable case, the sought after Finsler function is pro…

Based on a self-contained, coordinate-free exposition of the necessary concepts and tools of spray and Finsler geometry (with detailed proofs), we derive new results among others on the consequences of the direction-independence of the Landsberg tensor and the stretch tensor of a Finsler manifold. We show that an at le…

New sprays of constant curvature introduced; conditions for metrizability given.

The paper studies sprays on Hamel-Funk functions and their properties.

The Finsleroid--Finsler space becomes regular when the norm $||b||=c$ of the input 1-form $b$ is taken to be an arbitrary positive scalar $c(x) < 1$. By performing required direct evaluations, the respective spray coefficients have been obtained in a simple and transparent form. The adequate continuation into the regul…

Study left invariant spray structures on Lie groups, calculating curvature and geodesics.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

By performing required evaluations, we show that in the Finsleroid-regular space the Landsberg-space condition just degenerates to the Berwald-space condition (at any dimension number $N\ge2$). Simple and clear expository representations are obtained. Due comparisons with the Finsleroid-Finsler space are indicated. Key…

Geodesic sprays on Finsler manifolds studied with covariant coefficients.

The paper classifies Landsberg metrics on a 2D Lie group and proves a conjecture.

Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide new intrinsic (coordinate-free) proofs of intrinsic versions of the existence and uniqueness theorems for the Cartan and Berwald connections on a Finsler manifold. To accomplish this, the notions of semispray and nonli…

The pseudo-Finsleroid relativistic metric was constructed upon assuming that the involved vector field $b_i$ is time-like. In the present paper it is shown that the metric admits just the alternative counterpart in which the field is space-like. The entailed pseudo-Finsleroid-spatial framework is systematically describ…

The paper studies complex Finsler metrics on complex Lie groups.

The paper studies invariant functions and their relation to Landsberg surfaces.

The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical $\left( ρ,η\right)$-system with respect to a $(ρ, η)$-spray, the Berwald $(ρ, η)$-…

The paper studies anisotropic conformal changes in pseudo-Finsler surfaces.

The paper explores conditions for Finsler surfaces to be Landsbergian and classify surfaces with specific flag curvature conditions.

The paper explores parallel 1-forms on special Finsler manifolds and their properties.

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 differential system on the energy function can be reduced to a first order system on this …

Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group $S^3$. Using the Yasuda-Shimada theorem as an inspiration, we determine for each K>1 a privileged right invariant Killing field of constant length. Each such Riemannian metric p…

Study shows zero Pontrjagin classes for sprays.

Defines projective Ricci curvature and proves rigidity for sprays.

This paper classifies geodesics of projectively flat sprays and introduces a method to determine sprays based on geodesics.

Sprays with vanishing X-curvature are studied in this paper.

Geodesic orbit and weakly symmetric properties in spray geometry.

The paper studies Jacobi fields and conjugate points in projective sprays.

Sprays get Hamiltonian description using Dirac structures.

Sprays on Frechet manifolds connect connections and tangent structures.

Introduces submersion in spray geometry and defines key components.

The paper classifies spherically symmetric sprays and their curvature properties.

The abstract introduces golden Finsler structures and explores their local and global properties.

Study reformulates Finsler metrizability problems using geodesic invariance.

The main purposes of this article are to extend our previous results on homogeneous sprays to arbitrary (generalized) sprays, to show that locally diffeomorphic exponential maps can be defined for any (generalized) spray, and to give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. I…

Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.

The abstract proves properties of Berwald spaces with non-zero flag curvature.

Study left invariant spray geometry on Lie groups using parallel translations.

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 $1$. Moreover, these new metrics are not, generally, isometric to the Klein met…

The paper studies Berwald scalar curvature properties in Finsler geometry.

The pseudoconvex and disprisoning conditions for geodesics of linear connections are extended to the solution curves of general homogeneous sprays. The main result is that pseudoconvexity and disprisonment are jointly stable in the fine topology on the space of all homogeneous sprays of any degree of homogeneity.

In this work we show that for the geodesic spray $S$ of a Finsler function $F$ the most natural projective deformation $\widetilde{S}=S -2 λF\mathbb C$ leads to a non-Finsler metrizable spray, for almost every value of $λ\in \mathbb R$. This result shows how rigid is the metrizablility property with respect to certain …

After summarizing some necessary preliminaries and tools, including Berwald derivative and Lie derivative in pull-back formalism, we present ten equivalent conditions, each of which characterizes Berwald manifolds among Finsler manifolds. These range from Berwald's classical definition to the existence of a torsion-fre…

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 metrizability problem for a spray in terms of a first-order partial differential ope…

The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions f…

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 spray coefficients, we demonstrate how the Landsberg condition can explicitly appea…

This research examines anisotropic conformal transformations of pseudo-Finsler surfaces.

In this paper we are investigating variational homogeneous second order differential equations by considering the questions of how many different variational principles exist for a given spray. We focus our attention on h(2)-variationality; that is, the regular Lagrange function is homogeneous of degree two in the dire…

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 Finsler metrizable if and only if it is Riemann metrizable. This result means that…