On Randers spaces of constant flag curvature
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.
scientific article; zbMATH DE number 2053305
scientific article