Abstract
In this paper, we show that every homogeneous Finsler metric is a weakly stretch metric if and only if it reduces to a weakly Landsberg metric. This yields an extension of Tayebi–Najafi’s result that proved the result for the class of stretch Finsler metrics. Let \(F:=\alpha \phi (\beta /\alpha )\) be a homogeneous weakly stretch \((\alpha ,\beta )\)-metric on a manifold M. We show that if \(\phi \) is of polynomial type, then F is a Berwald metric. Also, we prove that F is a Berwald metric if and only if it has vanishing S-curvature. Then, we show that F is a Douglas metric if and only if it reduces to a Berwald metric. In continue, we show that every homogenous weakly stretch surface is a Landsberg surface. Finally, we characterize homogeneous weakly stretch spherically symmetric Finsler metrics.
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Acknowledgements
The authors would like to thank to Professor Akbar Tayebi for his encouragement and useful and exact comments during the preparation of this paper. Also, we are grateful to the anonymous referees for their suggestions which helped in improving the paper.
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Communicated by Mohammad Koushesh.
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Vishkaei, H.T., Toomanian, M., Katamy, R.C. et al. On Homogeneous Weakly Stretch Finsler Metrics. Bull. Iran. Math. Soc. 48, 19–30 (2022). https://doi.org/10.1007/s41980-020-00498-z
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DOI: https://doi.org/10.1007/s41980-020-00498-z
Keywords
- Weakly stretch metric
- Landsberg metric
- Weakly Landsberg metric
- Douglas metric
- Berwald metric
- Finsler surface
- Spherically symmetric Finsler metric