Foundations of singular Finsler geometry
type: Item
instance of: scholarly article
title: Foundations of singular Finsler geometry
zbMATH Open document ID: 1396.53101
author: Peng Zhang
published in: European Journal of Mathematics
publication date: 2018-02-08T00:00:00Z
review text: The author offers a singular (non-smooth) generalization of Finsler spaces. Let \(\left( X,d\right) \) be a non-reversible metric space (i.e., the symmetry \(d\left( x,y\right) =d\left( y,x\right) \) need not be true). Then oriented (geodesic) segments are defined via a positive isometry. The negative and positive variations of length, \(V_{-}\left( \gamma \right) \) and \(V_{+}\left( \gamma \right) \), are defined in a standard way with obvious adjustments for metric non-reversibility. A mapping \(\gamma \) is said to be rectifiable if \(d\left( \gamma \left( t\right) ,\gamma \left( t^{\prime }\right) \right) =d\left( \gamma \left( t^{\prime }\right) ,\gamma \left( t\right) \right) \) for all \(t\) and \(t^{\prime }\) and \(\gamma \) is both negatively and positively rectifiable. The space \(\left( X,d\right) \) is said to be geodesic if for every ordered pair of distinct points, there is a directed segment from \(x\) to \(y\). Let \(\gamma ,\gamma ^{\prime }\) be two positively oriented mappings (oriented geodesics) starting at the same point \(x_{0}\in X\). Then the oriented geodesics \(\gamma \) and \(\gamma ^{\prime }\) are said to have the same direction at \(x_{0}\) if both upper limits of \(d\left( \gamma \left( t\right) ,\gamma ^{\prime }\left( t\right) \right) /t\) and \(d\left( \gamma ^{\prime }\left( t\right) ,\gamma \left( t\right) \right) /t\) converge to zero as \(t\rightarrow 0+\). This equivalence relation on the set of all geodesics from \(x_{0}\) defines the set of directions. It is not difficult to introduce the upper isosceles angle between directions -- a straightforward substitute for an upper angle defined via half-angle. The upper isosceles angle defines a non-reversible metric on the set of all directions and determines the space of directions at \(x_{0}\). Finally, the author adopts the notion of a tangent cone (via Hausdorff convergence) for geodesic topologically symmetric (i.e., metric topologies \( \tau _{+}\)and \(\tau _{-}\) coincide…
reviewed by: Igor G. Nikolaev
Mathematics Subject Classification ID: 53C70
Mathematics Subject Classification ID: 53B40
Mathematics Subject Classification ID: 57N16
zbMATH DE Number: 6835951
zbMATH Keywords: singular Finsler space
zbMATH Keywords: directed segment
zbMATH Keywords: space of directions
zbMATH Keywords: tangent cone
zbMATH Keywords: curves with velocity
MaRDI profile type: Publication
full work available at URL: s40879-017-0169-x
OpenAlex ID: W2742778448
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DOI: 10.1007/S40879-017-0169-X
description: scientific article; zbMATH DE number 6835951