Abstract
In this paper, we define a weaker notion of Douglas metrics, namely weakly Douglas metrics. A Finsler metric satisfies a projectively invariant equation \(D^i_{\,\,jkl}=T_{jkl} y^i\) for some tensor \(T_{jkl}\) is called a weakly Douglas metric. We show that every Randers manifold of dimension \(n\ge 3\) is a weakly Douglas metric if and only if it is a Douglas metric. Then, we prove that every Kropina manifold is a weakly Douglas metric if and only if it is a Douglas metric. It turns out that every Kropina surface is a Douglas surface.
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Acknowledgements
The authors would like to thank Professors László Kozma and Professor Zhongmin Shen for their valuable comments and their encouragements during preparation of this manuscript.
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Atashafrouz, M., Najafi, B. & Tayebi, A. Weakly Douglas Finsler metrics. Period Math Hung 81, 194–200 (2020). https://doi.org/10.1007/s10998-020-00335-0
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DOI: https://doi.org/10.1007/s10998-020-00335-0