arXiv subjects

Csaba Vincze

Publications and source records attributed to Csaba Vincze.

At least 19 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.

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arXiv2025

An application of Brouwer's fixed-point theorem: continuously differentiable convex functions with gradient of constant norm

As an application of Brouwer's fixed-point theorem we prove that a continuously differentiable convex function with gradient of constant norm is an affine mapping. It is a first-order characterization of affine mappings among continuously differentiable convex functions, because neither the second-order condition of convexity nor related operators are used. The condition of differentiability is essential as the case of the norm function shows. In addition to Brouwer's theorem, the proof is based on the Cauchy--Bunyakovsky--Schwarz inequality and becomes complete by minimizing the distance between lines of gradient directions. Following the steps of the proof, we sketch a possible generalization of the result to functions defined on Hilbert spaces.

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arXiv2025

Randers metrics with compatible linear connections: a coordinate-free approach

A Randers space is a differentiable manifold equipped with a Randers metric. It is the sum of a Riemannian metric and a one-form on the base manifold. The compatibility of a linear connection with the metric means that the parallel transports preserve the Randers norm of tangent vectors. The existence of such a linear connection is not guaranteed in general. If it does exist then we speak about a generalized Berwald Randers metric. In what follows we give a necessary and sufficient condition for a Randers metric to be a generalized Berwald metric and we describe some distinguished compatible linear connections. The method is based on the solution of constrained optimization problems for tensors that are in one-to-one correspondence to the compatible linear connections. The solutions are given in terms of explicit formulas by choosing the free tensor components to be zero. Throughout the paper we use a coordinate-free approach to keep the geometric feature of the argumentation as far as possible.

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arXiv2025

On computable classes of equidistant sets: multivariate equidistant functions

An equidistant set in the Euclidean space consists of points having equal distances to both members of a given pair of sets, called focal sets. Having no effective formulas to compute the distance of a point and a set, it is hard to determine the points of an equidistant set in general. Special classes of equidistant sets allow us to approximate the equidistant points in more complicated cases. In the paper we have a hyperplane corresponding to the first order (linear) approximation for one of the focal sets and the second one is considered as the epigraph of a positive-valued continuous function. In the first part of the paper we prove that the equidistant points having equal distances to the epigraph of a positive-valued continuous function and its domain form the graph of a multivariate function. Therefore such an equidistant set is called a multivariate equidistant function. We also prove that the equidistant function one of whose focal sets is constituted by the pointwise minima of finitely many positive-valued continuous functions is given by the pointwise minima of the corresponding equidistant functions. In the second part of the paper we consider equidistant functions belonging to the epigraph of a convex function under some smoothness conditions. Independently of the dimension of the space we present a special parameterization for the equidistant points based on the closest point property of the epigraph as a convex set and we give the characterization of the equidistant functions as well. An example is also presented with a hyperboloid of revolution as one of the focal sets.

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arXiv2024

A geometric application of Lagrange multipliers: extremal compatible linear connections

The L\'evi-Civita connection of a Riemannian manifold is a metric (compatible) linear connection, uniquely determined by its vanishing torsion. It is extremal in the sense that it has minimal torsion at each point. We can extend this idea to more general spaces with more general (not necessarily quadratic) indicatrix hypersurfaces in the tangent spaces. Here, the existence of compatible linear connections on the base manifold is not guaranteed anymore, which needs to be addressed along with the intrinsic characterization of the extremal one. The first step is to provide the Riemann metrizability of the compatible linear connections. This Riemannian environment establishes a one-to-one correspondence between linear connections and their torsion tensors, also giving a way of measuring the length of the latter. The second step is to solve a hybrid conditional extremum problem at each point of the base manifold, all of whose constraint equations (compatibility equations) involve functions defined on the indicatrix hypersurface. The objective function to be minimized is a quadratic squared norm function defined on the finite dimensional fiber (vector space) of the torsion tensor bundle. We express the solution by using the method of Lagrange multipliers on function spaces point by point, we present a necessary and sufficient condition of the solvability and the solution is also given in terms of intrinsic quantities affecting the uniform size of the linear isometry groups of the indicatrices. This completes the description of differential geometric spaces admitting compatible linear connections on the base manifold, called generalized Berwald spaces, in Finsler geometry.

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arXiv2024

On locally symmetric polynomial metrics: Riemannian and Finslerian surfaces

In the paper we investigate locally symmetric polynomial metrics in special cases of Riemannian and Finslerian surfaces. The Riemannian case will be presented by a collection of basic results (regularity of second root metrics) and formulas up to Gauss curvature. In case of Finslerian surfaces we formulate necessary and sufficient conditions for a locally symmetric fourth root metric in 2D to be positive definite. They are given in terms of the coefficients of the polynomial metric to make checking the positive definiteness as simple and direct as possible. Explicit examples are also presented. The situation is more complicated in case of spaces of dimension more than two. Some necessary conditions and an explicit example are given for a positive definite locally symmetric polynomial metric in 3D. Computations are supported by the MAPLE mathematics software (LinearAlgebra).

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arXiv2023

On taxicab distance mean functions and their geometric applications: methods, implementations and examples

A distance mean function measures the average distance of points from the elements of a given set of points (focal set) in the space. The level sets of a distance mean function are called generalized conics. In case of infinite focal points the average distance is typically given by integration over the focal set. The paper contains a survey on the applications of taxicab distance mean functions and generalized conics' theory in geometric tomography: bisection of the focal set and reconstruction problems by coordinate X-rays. The theoretical results are illustrated by implementations in Maple, methods and examples as well.

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arXiv2022

Finsler metrics and semi-symmetric compatible linear connections

Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds. Compatible linear connections are the solutions of the so-called compatibility equations containing the torsion components as unknowns. Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics \cite{BM}, extremal compatible linear connections, algorithmic solutions \cite{V14}), it is very hard to solve in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved \cite{V10} (see also \cite{V11}) that such a compatible linear connection must be uniquely determined. The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. We present a new proof for the unicity result by using linear algebra and some basic facts about convex bodies and an explicit formula for the solution without integration. Necessary conditions of the solvability are also formulated in terms of intrinsic equations without unknown quantities. They are sufficient if and only if the solution depends only on the position.

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arXiv2021

On equidistant polytopes in the Euclidean space

An equidistant polytope is a special equidistant set in the space $\mathbb{R}^n$ all of whose boundary points have equal distances from two finite systems of points. Since one of the finite systems of the given points is required to be in the interior of the convex hull of the other one we can speak about inner and outer focal points of the equidistant polytope. It is of type $(q, p)$, where $q$ is the number of the outer focal points and $p$ is the number of the inner focal points. The equidistancy is the generalization of convexity because a convex polytope can be given as an equidistant polytope of type $(q, 1)$, where $q\geq n+1$. In the paper we present some general results about the basic properties of the equidistant polytopes: convex components, graph representations, connectedness, correspondence to the Voronoi decomposition of the space etc. Especially, we are interested in equidistant polytopes of dimension $2$ (equidistant polygons). Equidistant polygons of type $(3,2)$ will be characterized in terms of a constructive (ruler-and-compass) process to recognize them. In general they are pentagons with exactly two concave angles such that the vertices, where the concave angles appear at, are joined by an inner diagonal related to the adjacent sides of the polygon in a special way via the three reflection theorem for concurrent lines. The last section is devoted to some special arrangements of the focal points to get the concave quadrangles as equidistant polygons of type $(3,2)$.

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arXiv2021

On generalized Berwald manifolds of dimension three

A linear connection on a Finsler manifold is called compatible to the Finsler function if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. In the paper we present a general and intrinsic method to characterize the compatible linear connections on a Finsler manifold of dimension three. We prove that if a compatible linear connection is not unique then the indicatrices must be Euclidean surfaces of revolution. The surplus freedom of choosing compatible linear connections is related to Euclidean symmetries. The unicity of the solution of the compatibility equations can be provided by some additional requirements. Following the idea in \cite{V14} we are also looking for the so-called extremal compatible linear connection minimizing the norm of its torsion at each point of the manifold.

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arXiv2020

On the extremal compatible linear connection of a Randers space

A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the unicity of the linear connection in general, the first step of checking the existence of compatible linear connections on a Finsler manifold is to choose the best one to look for. A reasonable choice is introduced in \cite{V14} called the extremal compatible linear connection, which has torsion of minimal norm at each point. Randers metrics are special Finsler metrics that can be written as the sum of a Riemannian metric and a 1-form (they are "translates" of Riemannian metrics). In this paper, we investigate the compatibility equations for a linear connection to a Randers metric. Since a compatible linear connection is uniquely determined by its torsion, we transform the compatibility equations by taking the torsion components as variables. We determine when these equations have solutions, i.e. when the Randers space becomes a generalized Berwald space admitting a compatible linear connection. Describing all of them, we can select the extremal connection with the norm minimizing property. As a consequence, we obtain the characterization theorem in \cite{Vin1}: a Randers space is a non-Riemannian generalized Berwald space if and only if the norm of the perturbating term with respect to the Riemannian part of the metric is a positive constant.

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arXiv2019

On the extremal compatible linear connection of a generalized Berwald manifold

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibi\-li\-ty condition). By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is a strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. In the paper we consider the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is a conditional extremum problem involving functions defined on a local neighbourhood of the tangent manifold. In case of a given point of the manifold, the reference element method provides that the number of the Lagrange multipliers equals to the number of the equations providing the compatibility of the linear connection to the Finslerian metric. Therefore the solution of the conditional extremum problem with a reference element can be expressed in terms of the canonical data. The solution of the conditional extremum problem independently of the reference elements can be constructed algorithmically at each point of the manifold. The pointwise solutions constitute a section of the torsion tensor bundle for testing the compatibility of the corresponding linear connection to the Finslerian metric. In other words, we have an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.

math.DG↗

arXiv2019

On a class of linear functional equations without range condition

The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let $n\geq 2$ be an arbitrarily fixed integer, let further $X$ and $Y$ be linear spaces over the field $\mathbb{K}$ and let $α_{i}, β_{i}\in \mathbb{K}$, $i=1, \ldots, n$ be arbitrarily fixed constants. We will describe all those functions $f, f_{i, j}\colon X\times Y\to \mathbb{K}$, $i, j=1, \ldots, n$ that fulfill functional equation \[ f\left(\sum_{i=1}^n α_i x_i, \sum_{i=1}^n β_i y_i\right)= \sum_{i, j=1}^{n}f_{i, j}(x_i, y_j) \qquad \left(x_i \in X, y_i \in Y, i=1, \ldots, n\right). \] Additionally, necessary and sufficient conditions will also be given that guarantee the solutions to be non-trivial.

math.CA↗

arXiv2019

On compatible linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory \cite{V5} such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection is uniquely determined by its torsion tensor. If the torsion is zero then we have a classical Berwald manifolds. Otherwise the torsion is a strange data we need to express in terms of quantities of the Finsler manifold. In the paper we are going to give explicit formulas for the linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds. The results are based on averaging of (intrinsic) Finslerian quantities by integration over the indicatrix surfaces. They imply some consequences for the base manifold as a Riemannian space with respect to the averaged Riemannian metric. The possible cases are Riemannian spaces of constant zero curvature, constant positive curvature or Riemannian spaces admitting Killing vector fields of constant Riemannian length.

math.DG↗

arXiv2018

Characterization of field homomorphisms through Pexiderized functional equations

The aim of this paper is to prove characterization theorems for field homomorphisms. More precisely, the main result investigates the following problem. Let $n\in \mathbb{N}$ be arbitrary, $\mathbb{K}$ a field and $f_{1}, \ldots, f_{n}\colon \mathbb{K}\to \mathbb{C}$ additive functions. Suppose further that equation \[ \sum_{i=1}^{n}f^{q_{i}}_{i}\left(x^{p_{i}}\right)=0 \qquad \left(x\in \mathbb{K}\right) \] is also satisfied. Then the functions $f_{1}, \ldots, f_{n}$ are linear combinations of field homomorphisms from $\mathbb{K}$ to $\mathbb{C}$.

math.AC↗

arXiv2018

On functional equations characterizing derivations: methods and examples

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. The method allows us to refine the process of computing the solutions of univariate functional equations of the form \[ \sum_{k=1}^{n}x^{p_{k}}f_{k}(x^{q_{k}})=0, \] where $p_k$ and $q_k$ ($k=1, \ldots, n$) are given nonnegative integers and the unknown functions $f_{1}, \ldots, f_{n}\colon R\to R$ are supposed to be additive on the ring $R$. It is illustrated by some explicit examples too. As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that such a functional equation characterizes derivations of higher order. The results are uniformly based on the investigation of the multivariate version of the functional equations.

math.CA↗

arXiv2018

On convex closed planar curves as equidistant sets

The equidistant set of two nonempty subsets $K$ and $L$ in the Euclidean plane is a set all of whose points have the same distance from $K$ and $L$. Since the classical conics can be also given in this way, equidistant sets can be considered as a kind of their generalizations: $K$ and $L$ are called the focal sets. In their paper \cite{PS} the authors posed the problem of the characterization of closed subsets in the Euclidean plane that can be realized as the equidistant set of two connected disjoint closed sets. We prove that any convex closed planar curve can be given as an equidistant set, i.e. the set of equidistant curves contains the entire class of convex closed planar curves. In this sense the equidistancy is a generalization of the convexity.

math.MG↗

arXiv2018

On the generalization of Erdős-Vincze's theorem about the approximation of closed convex plane curves by polyellipses

A polyellipse is a curve in the Euclidean plane all of whose points have the same sum of distances from finitely many given points (focuses). The classical version of Erdős-Vincze's theorem states that regular triangles can not be presented as the Hausdorff limit of polyellipses even if the number of the focuses can be arbitrary large. In other words the topological closure of the set of polyellipses with respect to the Hausdorff distance does not contain any regular triangle and we have a negative answer to the problem posed by E. Vázsonyi (Weissfeld) about the approximation of closed convex plane curves by polyellipses. It is the additive version of the approximation of simple closed plane curves by polynomial lemniscates all of whose points have the same product of distances from finitely many given points (focuses). Here we are going to generalize the classical version of Erdős-Vincze's theorem for regular polygons in the plane. We will conclude that the error of the approximation tends to zero as the number of the vertices of the regular polygon tends to the infinity. The decreasing tendency of the approximation error gives the idea to construct curves in the topological closure of the set of polyellipses. If we use integration to compute the average distance of a point from a given (focal) set in the plane then the curves all of whose points have the same average distance from the focal set can be given as the Hausdorff limit of polyellipses corresponding to partial sums.

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