A class of Randers metrics of scalar flag curvature (Q2853390)

A Finsler metric \(F\) on a manifold \(M\) is called to be of weakly isotropic flag curvature if for the flag curvature \(K_{F}\) holds \(K_{F}=\frac{ 3\theta }{F}+{k}(x),\) where \(\theta =\theta _{i}(x)y^{i}\) is a \(1\)-form and \({k}(x)\) is a scalar function on \(M.\) NEWLINENEWLINENEWLINEIn this paper the authors investigate the Randers metric \(F=\alpha +\beta \) of scalar flag curvature (i.e., \(\alpha \) is flat and \(\beta \) is closed). More exactly, they study the Randers metric \(F=\alpha +\beta \) of projectively isotropic flag curvature. This means that there is a \(1\)-form \(\eta \) such that \(\bar{F}=\alpha +\bar{\beta},\) with \(\bar{\beta}:=\beta -\eta ,\) is of weakly isotropic flag curvature. In this case, \(F\) is projectively equivalent to \(\bar{F}\) and \(F\) is of scalar flag curvature. NEWLINENEWLINENEWLINEThe authors study the homothetic vector fields on Randers spaces of projectively isotropic flag curvature. By means of these and using the navigation idea, they construct new Randers metrics of scalar flag curvature.