On Randers spaces of constant flag curvature
type: Item
instance of: scholarly article
title: On Randers spaces of constant flag curvature
zbMATH Open document ID: 1048.53054
DOI: 10.1016/S0034-4877(03)80002-2
author: D. Bao
author: C. Robles
published in: Reports on Mathematical Physics
publication date: 2004-03-08T00:00:00Z
review text: The paper is concerned with a characterization of \(n\)-dimensional Randers spaces of constant flag curvature for \(n\geq3\), or whose flag curvature is a function of the position only for \(n=2\). The known classifications of Randers spaces of constant flag curvature [\textit{M. Matsumoto}, Rep. Math. Phys. 28, 249--261 (1989; Zbl 0734.53022); \textit{C. Shibata} and \textit{M. Kitayama}, Proc. Rom.-Jap. Colloq., 139--157 (1985; Zbl 0647.53014); \textit{H. Yasuda} and \textit{H. Shimada}, Rep. Math. Phys. 11, 347--360 (1977; Zbl 0362.53051)] all claim that a particular tensor, here denoted \(\text{curl}_{\widetilde{b}i}\), must vanish if a Randers metric is to have constant flag curvature. Contrary to that claim, the authors give a variety of examples of Randers spaces of constant flag curvature with non-vanishing \(\text{curl}_{\widetilde{b}i}\). This implies that the Yasuda-Shimada classification of Randers space forms addresses only the special case \(\text{curl}_{\widetilde{b}i}=0\). The authors investigate three necessary and sufficient conditions for a Randers space to have constant flag curvature. They apply their characterization to the special case \(\text{curl}_{\widetilde{b}i}=0\) and prove that the corrected classification for that special case is sharp, holds for all \(n\geq2\), and follows directly from the given three necessary and sufficient conditions.
Mathematics Subject Classification ID: 53C60
Mathematics Subject Classification ID: 53B40
zbMATH DE Number: 2053305
zbMATH Keywords: flag curvature
zbMATH Keywords: spray curvature
zbMATH Keywords: Berwald's formula
zbMATH Keywords: Randers space
zbMATH Keywords: Yasuda-Shimada theorem
reviewed by: Nabil L. Youssef
MaRDI profile type: Publication
cites work: Q4204736
cites work: The theory of sprays and Finsler spaces with applications in physics and biology
cites work: Q3128806
cites work: Q4661850
cites work: Zermelo navigation on Riemannian manifolds
cites work: Finsler Metrics of Constant Positive Curvature on the Lie Group S 3
cites work: Über Finslersche und Cartansche Geometrie. IV. Projektivkrümmung allgemeiner affiner Räume und Finslerscher Räume skalarer Krümmung
cites work: Randers spaces of constant curvature
cites work: Q4173917
cites work: Q3223491
cites work: On an Asymmetrical Metric in the Four-Space of General Relativity
cites work: Q3260127
cites work: Differential geometry of spray and Finsler spaces
cites work: Finsler Metrics with K = 0 and S = 0
cites work: Two-dimensional Finsler metrics with constant flag curvature
cites work: Q3791804
cites work: Q4139285
cites work: On Randers spaces of scalar curvature
cites work: Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung
Wikidata QID: Q126203515
description: scientific article; zbMATH DE number 2053305