## 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.
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