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On Some Explicit Constructions of Finsler Metrics with Scalar Flag Curvature

  • Xiaohuan Mo (a1) and Changtao Yu (a2)

Abstract

We give an explicit construction of polynomial (of arbitrary degree) $(\alpha ,\,\beta )$ -metrics with scalar flag curvature and determine their scalar flag curvature. These Finsler metrics contain all nontrivial projectively flat $(\alpha ,\,\beta )$ -metrics of constant flag curvature.

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Corresponding author

Footnotes

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This work is supported by the National Natural Science Foundation of China 10471001.

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References

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[1] Berwald, L., Untersuchung der Krümmung allgemeiner metrischer Räume auf Grund des in ihnen herrschenden Parallelismus. Math. Z. 25(1926), 40–73. doi:10.1007/BF01283825
[2] Berwald, L., Parallelübertragung in allgemeinen Räumen. Atti Congr. Intern. Mat. Bologna 4(1928), 263–270.
[3] Chen, X., Mo, X. and Shen, Z., On the flag curvature of Finsler metrics of scalar curvature. J. London Math. Soc. 68(2003), 762–780. doi:10.1112/S0024610703004599
[4] Chern, S. S. and Shen, Z., Riemann–Finsler geometry. Nankai Tracts in Mathematics 6,World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.
[5] Li, B. and Shen, Z., On a class of projectively flat Finsler metrics with constant flag curvature. Internat. J. Math. 18(2007), 749–760. doi:10.1142/S0129167X07004291
[6] Mo, X., A global classification result for Randers metrics of scalar curvature on closed manifolds. Nonlinear Analysis, 69(2008), 2996-3004. doi:10.1016/j.na.2007.08.067
[7] Mo, X. and Shen, Z., On negatively curved Finsler manifolds of scalar curvature. Canad. Math. Bull. 48(2005), 112–120.
[8] Mo, X., Shen, Z. and Yang, C., Some constructions of projectively flat Finsler metrics. Sci. China Ser. A 49(2006), 703–714. doi:10.1007/s11425-006-0703-7
[9] Shen, Z., Differential Geometry of Spray and Finsler Spaces. Kluwer Academic Publishers, 2001.
[10] Shen, Z., Projectively flat Randers metrics with constant flag curvature. Math. Ann. 325(2003), 19–30. doi:10.1007/s00208-002-0361-1
[11] Shen, Z., On some projectively flat (Finsler) metrics. http://www.math.iupui.edu/»zshen/Research/papers/, 2005.
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On Some Explicit Constructions of Finsler Metrics with Scalar Flag Curvature

  • Xiaohuan Mo (a1) and Changtao Yu (a2)

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