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Projectively flat exponential Finsler metric. (English) Zbl 1104.53019

On an open domain of \(\mathbb R^n\) let be given a Riemannian metric \(\alpha \) and a 1-form \(\beta \). For a constant \(\varepsilon \) the function \(F=\alpha \exp(\frac{\beta }{\alpha }+\varepsilon \beta)\) is a Finsler metric called exponential. The main result: \(F\) is locally projectively flat if and only if \(\alpha \) is projectivelly flat and \(\beta \) is parallel with respect to \(\alpha \). Also, it is proved that the Douglas tensor of \(F\) vanishes if and only if \(\beta \) is parallel with respect to \(\alpha \). As consequence, an exponential Finsler metric of Douglas type is a Landsberg metric.
Reviewer: Radu Miron (Iaşi)

MSC:

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