On \((\alpha , \beta )\)-metrics of scalar flag curvature with constant S-curvature (Q606325)

Flag curvature is the natural generalization of sectional curvature in Riemannian geometry to Finsler geometry. One of the central problems in Finsler geometry is to study flag curvature and to characterize Finsler spaces with special properties in terms of flag curvature. A Finsler space is called of scalar curvature if its flag curvature is a function of the flagpole only. In this paper, the author studies a special type of Finsler metrics, namely \((\alpha,\beta)\)-metrics, with scalar flag curvature and constant S-curvature. The main result is that an \((\alpha,\beta)\)-metric (not of Randers type) of dimension \(\geq 3\) is of scalar flag curvature with vanishing S-curvature if and only if it is a locally Minkowski metric.