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Cs. Vincze

Publications and source records attributed to Cs. Vincze.

4 recordsLinked to original sources
arXiv2026

On a special class of equidistant sets in the Euclidean space

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. Since there is no effective formula to compute the distance of a point and a set, it is hard to determine the points of an equidistant set in general. Therefore, it is important to investigate some special cases. In the paper we investigate equidistant sets that can be given as the graph of a function. They are called equidistant functions. In the previously examined conceptual model, one of the focal sets is the horizontal hyperplane through the origin and the other one is the epigraph of a positive-valued, continuous function. The equidistant points form the graph of another function over the hyperplane. In a general situation, the hyperplane is the first-order (linear) approximation for one of the focal sets. A natural idea is to substitute the hyperplane by a circle (sphere) as a second-order (quadratic) approximation for one of the focal sets in more complicated cases. Such a generalization results in a new type of equidistant functions we are going to investigate in the present paper. Before considering the special cases in detail, we present some general observations: a necessary and sufficient condition for the existence of equidistant points along the vertical lines, upper/lower equidistant functions, equidistant functions, a necessary and sufficient condition for the existence of the equidistant function, equidistant functions and the minimum operator (a kind of commuting property).

math.MG↗
arXiv2020

On the divergence representation of the Gauss curvature of Riemannian surfaces and its applications

In the paper we consider Riemannian surfaces admitting a global expression of the Gauss curvature as the divergence of a vector field. It is equivalent to the existence of a metric linear connection of zero curvature. Such a linear connection $\nabla$ plays an important role in the differential geometry of non-Riemannian surfaces in the sense that the Riemannian quadratic forms can be changed into Minkowski functionals in the tangent planes such that the Minkowskian length of the tangent vectors is invariant under the parallel translation with respect to $\nabla$ (compatibility condition). A smoothly varying family of Minkowski functionals in the tangent planes is called a Finslerian metric function under some regularity conditions. Especially, the existence of a compatible linear connection provides the Finsler surface to be a so-called generalized Berwald surface. It is an alternative of the Riemannian geometry for $\nabla$. Using some general observations and topological obstructions we concentrate on explicit examples. In some representative cases (Euclidean plane, hyperbolic plane etc.) we solve the differential equation of the parallel vector fields to construct a smoothly varying family of Minkowski functionals in the tangent planes such that the Minkowskian length of the tangent vectors is invariant under the parallel translation.

math.DG↗
arXiv2018

On generalized Berwald surfaces with locally symmetric fourth root metrics

Let $m=2l$ be a positive natural number, $l=1, 2, \ldots. $ A Finslerian metric $F$ is called an $m$-th root metric if its $m$-th power $F^m$ is of class $C^{m}$ on the tangent manifold $TM$. Using some homogenity properties, the local expression of an $m$-th root metric is a polynomial of degree $m$ in the variables $y^1$, $\ldots$, $y^n$, where $\dim M=n$. $F$ is locally symmetric if each point has a coordinate neighbourhood such that $F^m$ is a symmetric polynomial of degree $m$ in the variables $y^1$, $\ldots$, $y^n$ of the induced coordinate system on the tangent manifold. Using the fundamental theorem of symmetric polynomials, the reduction of the number of the coefficients depending on the position makes the computational processes more effective and simple. In the paper we present some general observations about locally symmetric $m$-th root metrics. Especially, we are interested in generalized Berwald surfaces with locally symmetric fourth root metrics. The main result (Theorem 1) is their intrinsic characterization in terms of the basic notions of linear algebra. We present a one-parameter family of examples as well. The last section contains some computations in 3D. They are supported by the MAPLE mathematics softwer (LinearAlgebra).

math.GM↗