Abstract
We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classical projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from the Weyl projective curvature, it is not a projective invariant, and hence Beltrami Theorem does not work in Finsler geometry. We provide the relation between the Weyl-type curvature tensors of two projectively related Finsler metrics. Using this formula we show that a projective deformation preserves the property of having constant flag curvature if and only if the projective factor is a Hamel function. This way we provide a Finslerian version of Beltrami Theorem.
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Acknowledgements
We express our thanks to Vladimir Matveev, Zhongmin Shen and József Szilasi for their comments and suggestions on this work. This work is supported by Ministry of Research and Innovation within Program 1—Development of the national RD system, Subprogram 1.2— Institutional Performance—RDI excellence funding projects, Contract no. 34PFE/19.10.2018.
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Bucataru, I., Creţu, G. A Characterisation for Finsler Metrics of Constant Curvature and a Finslerian Version of Beltrami Theorem. J Geom Anal 30, 617–631 (2020). https://doi.org/10.1007/s12220-019-00158-7
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DOI: https://doi.org/10.1007/s12220-019-00158-7