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On Matsumoto-type Finsler metrics. (English) Zbl 1276.53028

Authors’ abstract: In this paper, we find a necessary and sufficient condition for the Matsumoto-type metric \(F=\frac{(\alpha-\beta)^q}{\alpha^{q-1}}\) to be projectively flat. Suppose that the Matsumoto-type metric \(\overline{F}=\frac{(\alpha-\beta)^q}{\alpha^{q-1}}\) is constructed from a Randers metric \(F=\alpha+\beta\) on a manifold \(M\) by a \(\lambda\)-deformation, where \(\lambda\in C^\infty(M)\). We find a condition on \(\lambda\) under which the corresponding \(\lambda\)-deformation preserves the property of being projectively flat. We show that if \(\overline{F}=\frac{\lambda\alpha^2}{\alpha-\beta}\) is a \(\lambda\)-deformation of a Randers metric \(F=\alpha+\beta\), then \(F\) is a Berwald metric if and only if \(\overline{F}\) is a Douglas metric if and only if \(F\) is projectively related to \(\overline{F}\).

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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