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Hilbert’s Fourth Problem and Projectively Flat Finsler Metrics

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Lie Groups, Differential Equations, and Geometry

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Abstract

This survey article mainly introduces some important research progresses on the smooth solutions of Hilbert’s fourth problem in the regular case, which we call projectively flat Finsler metrics. We characterize and classify projectively flat Finsler metrics of constant flag curvature. We also discuss and classify projectively flat Finsler metrics with isotropic S-curvature. In particular, we study and characterize projectively flat Randers metrics, square metrics, (α, β)-metrics, and their curvature properties.

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References

  1.  J.C. Álvarez Paiva, Symplectic geometry and Hilbert’s fourth problem. J. Differ. Geom. 69(2), 353–378 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Bácsó, M. Matsumoto, On Finsler spaces of Douglas type. A generalization of the notion of Berwald space. Publ. Math. Debr. 51, 385–406 (1997)

    MATH  Google Scholar 

  3. D. Bao, C. Robles, On Randers metrics of constant curvature. Rep. Math. Phys. 51, 9–42 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Berwald, Über die n-dimansionalen Geometrien konstanter Krümmung, in denen die Geraden die kürzesten sind. Math. Z. 30, 499–469 (1929)

    Article  MATH  Google Scholar 

  5. R. Bryant, Finsler structures on the 2-sphere satisfying K = 1, in Finsler Geometry, Contemporary Mathematics, vol. 196 (American Mathematical Society, Providence, 1996), pp. 27–42

    Google Scholar 

  6. R. Bryant, Projectively flat Finsler 2-spheres of constant curvature. Sel. Math. 3, 161–204 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. R. Bryant, Some remarks on Finsler manifolds with constant flag curvature. Houst. J. Math. 28(2), 221–262 (2002)

    MathSciNet  MATH  Google Scholar 

  8. H. Busemann, Problem IV: Desarguesian spaces in mathematical developments arising from Hilbert problems, in Proceedings of Symposia in Pure Mathematics (American Mathematical Society, Providence, 1976)

    Google Scholar 

  9. X. Cheng, On (α, β)-metrics of scalar flag curvature with constant S-curvature. Acta Math. Sin. 26(9), 1701–1708 (2010)

    Google Scholar 

  10. X. Cheng, The (α, β)-metrics of scalar flag curvature. Differ. Geom. Appl. 35, 361–369 (2014)

    Google Scholar 

  11. X. Cheng, Z. Shen, Projectively flat Finsler metrics with almost isotropic S-curvature. Acta Math. Sci. 26B(2), 307–313 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. X. Cheng, Z. Shen, A class of Finsler metrics with isotropic S-curvature. Isr. J. Math. 169(1), 317–340 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. X. Cheng, Z. Shen, Finsler Geometry—An Approach via Randers Spaces (Springer/Science Press, Berlin, 2012)

    MATH  Google Scholar 

  14. X. Cheng, X. Mo, Z. Shen, On the flag curvature of Finsler metrics of scalar curvature. J. Lond. Math. Soc. 68(2), 762–780 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. S.S. Chern, Z. Shen, Riemann-Finsler Geometry. Nankai Tracts in Mathematics, vol. 6 (World Scientific, Singapore, 2005)

    Google Scholar 

  16. P. Funk, Über Geometrien bei denen die Geraden die kürzesten sind. Math. Ann. 101, 226–237 (1929)

    Article  MathSciNet  MATH  Google Scholar 

  17. G. Hamel, Über die Geometrien in denen die Geraden die kürzesten sind. Math. Ann. 57, 231–264 (1903)

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Hilbert, Mathematical problems. Bull. Am. Math. Soc. 8, 437–479 (1902); Bull. Am. Math. Soc. 37(4), 407–436 (2000)

    Google Scholar 

  19. B. Li, Z. Shen, On a class of projectively flat Finsler metrics with constant flag curvature. Int. J. Math. 18(7), 1–12 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Matsumoto, On C-reducible Finsler spaces. Tensor N. S. 24, 29–37 (1972)

    MathSciNet  MATH  Google Scholar 

  21. A.V. Pogorelov, Hilbert’s Fourth Problem. Scripta Series in Mathematics (Winston and Sons, Washington, 1979)

    Google Scholar 

  22. G. Randers, On an asymmetrical metric in the four-space of general relativity. Phys. Rev. 59, 195–199 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  23. Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry. Adv. Math. 128(2), 306–328 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Z. Shen, Finsler metrics with K=0 and S=0. Can. J. Math. 55(1), 112–132 (2003)

    Article  MATH  Google Scholar 

  25. Z. Shen, Projectively flat Randers metrics with constant flag curvature. Math. Ann. 325, 19–30 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Z. Shen, Projectively flat Finsler metrics of constant flag curvature. Trans. Am. Math. Soc. 355(4), 1713–1728 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Z. Shen, On projectively flat (α, β)-metrics. Can. Math. Bull. 52(1), 132–144 (2009)

    Google Scholar 

  28. Z. Shen, G.C. Yildirim, On a class of projectively flat metrics with constant flag curvature. Can. J. Math. 60(2), 443–456 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. C. Yu, Deformations and Hilbert’s fourth problem. Math. Ann. 365, 1379–1408 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. L. Zhou, A local classification of a class of (α, β) metrics with constant flag curvature. Differ. Geom. Appl. 28, 170–193 (2010)

    Google Scholar 

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Acknowledgements

X. Cheng was supported by the National Natural Science Foundation of China (11371386) and the European Union’s Seventh Framework Programme (FP7/2007–2013) under grant agreement no. 317721.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing University of Technology, Chongqing, 400054, People’s Republic of China

    Xinyue Cheng, Xiaoyu Ma, Yuling Shen & Shuhua Liu

Authors
  1. Xinyue Cheng
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  2. Xiaoyu Ma
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  3. Yuling Shen
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  4. Shuhua Liu
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Corresponding author

Correspondence to Xinyue Cheng .

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Editors and Affiliations

  1. Dipartimento di Matematica e Informatica, University of Palermo, Palermo, Italy

    Giovanni Falcone

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Cheng, X., Ma, X., Shen, Y., Liu, S. (2017). Hilbert’s Fourth Problem and Projectively Flat Finsler Metrics. In: Falcone, G. (eds) Lie Groups, Differential Equations, and Geometry. UNIPA Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-319-62181-4_11

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