arXiv research

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 to decide whether or not a given homogeneous system of second order ordinary dif…

The metrizability problem for a symmetric affine connection on a manifold, invariant with respect to a group of diffeomorphisms G, is considered. We say that the connection is G-metrizable, if it is expressible as the Levi-Civita connection of a G-invariant metric field. In this paper we analyze the G-metrizability equ…

Investigates projective Finsler metrizability for non-isotropic SODEs.

Study solutions to metrizability equation on manifolds.

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 condition, together with three other conditions on the Jacobi endomorphism, chara…

Study of quasi-Kähler metrics on complex manifolds linked to c-projective metrizability.

PhD dissertation on Finsler geometry and gravity, focusing on Berwald spaces and exact solutions.

We use the Frölicher-Nijenhuis formalism to reformulate the inverse problem of the calculus of variations for a system of differential equations of order 2k in terms of a semi-basic 1-form of order k. Within this general context, we use the homogeneity proposed by Crampin and Saunders in [14] to formulate and discuss t…

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 …

An invariant of three-dimensional orientable manifolds is built on the base of a solution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.

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…

In [17] A. Rapcsák obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential equations. In this paper we investigate the integrability of the Rapcsák system, consisting of the Rapcsák equations and the homogeneity condition, by using the Spence…

Study of non-metrizable manifolds' ends, generalizing Nyikos's theorem.

3D projective structures can be metrized with conformal structures.

The paper explores non-metrizability of projective deformations of Finsler sprays.

Several authors have pointed out the connection between Barbilian's metric introduced in 1934 and the recent study of Apollonian metrics. We provide examples of various distances that can be obtained by Barbilian's metrization procedure and we discuss the relation between this metrization procedure and important Rieman…

New insights into metrizability of SO(3)-invariant connections, linking Riemann and Finsler structures.

Study on metrizability and Ricci-flatness of Finsler spaces with Kropina metrics.

A linear connection $D$ in a Lie algebroid is said to be metrizable if there exists a Riemannian metric $h$ in the Lie algebroid such that $Dh=0$. Conditions for the linear connection $D$ to be metrizable are investigated.

New method for creating metrics in complex geometries.

The paper shows how MMD metrizes weak convergence for certain kernels.

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

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…

Study orbit spaces of equivariant ANEs for proper actions of metrizable groups.

Study investigates metrizability of Finsler spaces with specific metrics.

Short note explores conditions for Finsler functions to be uniquely metrizable.

Study reformulates Finsler metrizability problems using geodesic invariance.

Three themes of general topology: quotient spaces; absolute retracts; and inverse limits - are reapproached here in the setting of metrizable uniform spaces, with an eye to applications in geometric and algebraic topology. The results include: 1) If f: A -> Y is a uniformly continuous map, where X and Y are metric spac…

Quantum bundles use metrics to measure distances.

Endowed with quotient topology inherited from the space of based loops, the fundamental group of the Hawaiian earring fails to be metrizable. The fundamental group of any space which retracts to the Hawaiian earring is also nonmetrizable.

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 …

A new compact metrizable space Z replaces X in extension theory.

The paper proves metrizability and dynamics of Weil bundles.

Two new classes of metrizable vector bundles have been presented in the papers [1] and [4]. The Lie algebroid generalized tangent bundle of a dual vector bundle is presented. This Lie algebroid is a new example of metrizable vector bundle. A new class of Hamilton spaces, called by use, generalized Hamilton (ρ,η)-space,…

A class of metrizable vector bundles in the general framework of generalized Lie algebroids have been presented in the eight reference. Using a generalized Lie algebroid we obtain the Lie algebroid generalized tangent bundle of a vector bundle. This Lie algebroid is a new example of metrizable vector bundle. A new clas…

Fine shape of local compacta represented by ordinary maps.

Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.

Fine shape theory extends strong shape to noncompact metrizable spaces.

We investigate the problem of describing the homotopy classes $[X,Y]$ of continuous functions between $ω$-bounded non metrizable manifolds $X,Y$. We define a family of surfaces $X$ built with the first octant $C$ in $L^2$ ($L$ is the longline and $R$ the longray), and show that $[X,R]$ is in bijection with so called `a…

The paper examines parallel one forms on Riemannian and Finslerian manifolds.

New findings show Berwald Finsler spacetimes cannot be metrized.

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

We say that a metrizable space $M$ is a Krasinkiewicz space if any map from a metrizable compactum $X$ into $M$ can be approximated by Krasinkiewicz maps (a map $g\colon X\to M$ is Krasinkiewicz provided every continuum in $X$ is either contained in a fiber of $g$ or contains a component of a fiber of $g$). In this pap…

The Bergman measure converges to the Zhang measure on a hybrid space.

Study chaotic behavior in homeomorphism groups of countable products of spaces.

Introduces fine shape theory to simplify shape and antishape invariants.

The paper introduces a new spray deformation and finds new metrics.

We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which $\sqcup_{g} M_{g}$ is open dense. The metrized graphs, which are oft…