Searcharxiv⌕ Search

arXiv subjects

Zoltan Muzsnay

Publications and source records attributed to Zoltan Muzsnay.

16 recordsLinked to original sources
arXiv2026

Natural parallel translation and connection associated to navigation data

In this paper, we consider the geometric setting of navigation data and introduce a natural parallel translation using the Riemannian parallelism. The geometry obtained in this way has some nice and natural features: the natural parallel translation is homogeneous (but in general nonlinear), preserves the Randers type Finslerian norm constituted by the navigation data, and the holonomy group is finite-dimensional.

math.DG↗
arXiv2024

The geometry of geodesic invariant functions and applications to Landsberg surfaces

In this paper, for a given spray $S$ on an $n$-dimensional manifold $M$, we investigate the geometry of $S$-invariant functions. For an $S$-invariant function $¶$, we associate a vertical subdistribution $\V_¶$ and find the relation between the holonomy distribution and $\V_¶$ by showing that the vertical part of the holonomy distribution is the intersection of \ok{all spaces $\V_{\F_S}$ associated to $\F_S$ where $\F_S$} is the set of all Finsler functions that have the geodesic spray $S$. As an application, we study the Landsberg Finsler surfaces. We prove that a Landsberg surface with $S$-invariant flag curvature is Riemannian or has a vanishing flag curvature. We show 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 prove that the flag curvature is $H$-invariant if and only if it is constant.

math.DG↗
arXiv2024

The holonomy of spherically symmetric projective Finsler metrics of constant curvature

In this paper, we investigate the holonomy group of $n$-dimensional projective Finsler metrics of constant curvature. We establish that in the spherically symmetric case, the holonomy group is maximal, and for a simply connected manifold it is isomorphic to $Diff_o({\mathbb S^{n-1}})$, the connected component of the identity of the group of smooth diffeomorphism on the $(n-1)$-dimensional sphere. In particular, the holonomy group of the n-dimensional standard Funk metric and the Bryant-Shen metrics are maximal and isomorphic to $Diff_o({\mathbb S^{n-1}})$. These results are the firsts describing explicitly the holonomy group of n-dimensional Finsler manifolds in the non-Berwaldian (that is when the canonical connection is non-linear) case.

math.DG↗
arXiv2019

Metrizability of holonomy invariant projective deformation of sprays

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 the projective deformation $\widetilde{S}=S-2λP C$. We prove that for any holonomy invariant nontrivial function $P$ and for almost every value $λ\in R$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the holonomy invariant function is necessarily one of the principal curvatures of the geodesic structure.

math.DG↗
arXiv2019

Tangent Lie algebra of a diffeomorphism group and application to holonomy theory

In this paper we introduce the notion of tangent space TG of a (not necessary smooth) subgroup G of the diffeomorphism group Diff(M) of a compact manifold M. We prove that TG is a Lie subalgebra of the Lie algebra of smooth vector fields on M. The construction can be generalized to subgroups of any (finite or infinite dimensional) Lie groups. The tangent Lie algebra TG introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example, we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.

math.DG↗
arXiv2018

The holonomy group of projectively flat Randers two-manifolds of constant curvature

In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of the circle.

math.DG↗
arXiv2012

Tangent Lie algebras to the holonomy group of a Finsler manifold

Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesimal holonomy algebra by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. At the end we introduce the notion of the holonomy algebra of a Finsler manifold by all conjugates of infinitesimal holonomy algebras by parallel translations with respect to the Berwald connection. We prove that this holonomy algebra is tangent to the holonomy group.

math.DG↗
arXiv2012

Finsler 2-manifolds whose holonomy group is the diffeomorphism group of the circle

In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant-Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian holonomy structures.

math.DG↗
arXiv2012

Projectively flat Finsler manifolds with infinite dimensional holonomy

Recently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group can not be a compact Lie group, if the Finsler manifold of dimension $> 2$ has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant-Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group.

math.DG↗
arXiv2010

Finsler spaces with infinite dimensional holonomy group

Our paper is devoted to the study of the holonomy groups of Finsler surfaces using the methods of infinite dimensional Lie theory. The notion of infinitesimal holonomy algebra will be introduced, by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. We obtain that the topological closure of the holonomy group contains the exponential image of any tangent Lie algebra of the holonomy group. A class of Randers surfaces is determined, for which the infinitesimal holonomy algebra coincides with the curvature algebra. We prove that for all projectively flat Randers surfaces of non-zero constant flag curvature the infinitesimal holonomy algebra has infinite dimension and hence the holonomy group cannot be a Lie group of finite dimension. Finally, in the case of the Funk metric we prove that the infinitesimal holonomy algebra is a dense subalgebra of the Lie algebra of the full diffeomorphism group and hence the topological closure of the holonomy group is the orientation preserving diffeomorphism group of the circle.

math.DG↗
arXiv2010

Finsler manifolds with non-Riemannian holonomy

The aim of this paper is to show that holonomy properties of Finsler manifolds can be very different from those of Riemannian manifolds. We prove that the holonomy group of a positive definite non-Riemannian Finsler manifold of non-zero constant curvature with dimension >2 cannot be a compact Lie group. Hence this holonomy group does not occur as the holonomy group of any Riemannian manifold. In addition, we provide an example of left invariant Finsler metric on the Heisenberg group, so that its holonomy group is not a (finite dimensional) Lie group. These results give a positive answer to the following problem formulated by S. S. Chern and Z. Shen: "Is there a Finsler manifold whose holonomy group is not the holonomy group of any Riemannian manifold?"

math.DG↗
arXiv2006

Linearizable 3-webs and the Gronwall conjecture

In the article "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp.2643-2654), published in 2001, we studied the linearizability problem for 3-webs on a 2-dimensional manifold. Four years after the publication of our article, V.V.Goldberg and V.V.Lychagin in the paper "On linearization of planar three-webs and Blaschke's conjecture" (C.R.Acad. Sci. Paris, Ser. I. vol. 341. num 3 (2005)) obtained similar results by a different method and criticized our article by qualifying the proofs incomplete. However, they obtained false result on the linearizability of a certain web. We present here the complete version of our work with computations and explicit formulas, because we deem that their opinion concerning our work is unjustified.

math.DG↗
arXiv2006

On the problem of linearizability of a 3-web

In this paper we study the linearizability problem for 3-webs on a 2-dimensional manifold. With an explicit computation based on the theory developed in the paper "On the linearizability of 3-webs" (Nonlinear analysis 47, (2001) pp. 2643-2654), we examine a 3-web whose linearizability was claimed in the same paper. We show that, contrary to the statement of the papers "On the Blaschke conjecture for 3-webs" (arXiv: math.DG/0411460) and "On linearization of planar three-webs and Blaschke's conjecture", (C.R.Acad. Sci. Paris, Ser. I. vol. 341. num 3 (2005)), this particular web is linearizable. We compute explicitly the affine deformation tensor and the corresponding flat linear connection adapted to the web which linearizes it.

math.DG↗
arXiv2006

The Euler-Lagrange PDE and Finsler metrizability

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 same function. In this way we are able to give effective necessary and sufficient conditions for the local existence of a such Finsler metric in terms of the holonomy algebra generated by horizontal vector-fields. We also consider the Landsberg metrizability problem and prove similar results. This reduction is a significant step in solving the problem whether or not there exists a non-Berwald Landsberg space.

math.DG↗