On projectively flat \((\alpha, \beta)\)-metrics. (English) Zbl 1170.53012
The author studies a particular case of Finsler metrics, so-called \((\alpha, \beta)\)-metrics (i.e. of the form \(F=\alpha\phi(\beta/\alpha)\) where \(\alpha\) is a Riemann metric and \(\beta\) is a 1-form). It is proven that such a metric – under some conditions – is projectively flat (namely, geodesics are straight lines) on an open set if and only if \(\phi\), \(\beta\) and \(\alpha\) satisfy some conditions described nicely in the paper. This result generalizes the case where \(\phi(s)=1+s\) (i.e. a Randers metric) (see, e.g., [S. -S. Chern and Z. Shen , Riemann-Finsler geometry. Nankai Tracts in Mathematics 6. (Hackensack), NJ: World Scientific. (2005; Zbl 1085.53066)]).
Reviewer: Sofiane Bouarroudj (Al-Ain)
MSC:
53B40 | Local differential geometry of Finsler spaces and generalizations (areal metrics) |
53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |