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On Randers metrics with isotropic S-curvature

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Abstract

In this paper, we study a class of Finsler metrics defined by a vector field on a Riemannian space form. We give an explicit formula for those with isotropic S-curvature. This class contains all Randers metrics of constant flag curvature.

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References

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

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Mo, X.: On the flag curvature of a Finsler space with constant S-curvature. Houston J. Math., 31, 131–141 (2005)

    MATH  MathSciNet  Google Scholar 

  4. Shen, Z.: Finsler manifolds with nonpositive flag curvature and constant S-curvature. Math. Z., 249, 625–639 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Shen, Z.: Finsler metrics with K = 0 and S = 0. Canadian J. Math., 55, 112–132 (2003)

    MATH  Google Scholar 

  6. Shen, Z.: Two-dimensional Finsler metrics of constant curvature. Manu. Math., 109(3), 349–366 (2002)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, X., Shen, Z.: Randers metrics with special curvature properties. Osaka J. of Math., 40, 87–101 (2003)

    MATH  MathSciNet  Google Scholar 

  9. Xing, H.: The geometric meaning of Randers metrics with isotropic S-curvature. Adv. in Math. (Chinese), 34, 717–730 (2005)

    MathSciNet  Google Scholar 

  10. Bao, D., Robles, C., Shen, Z.: Zermelo Navigation on Riemannian manifolds. J. Diff. Geom., 66, 391–449 (2004)

    MathSciNet  Google Scholar 

  11. Cheng, X., Shen, Z.: Randers metrics of scalar flag curvature, preprint, 2005

  12. Eisenhart, L. P.: Riemannian geometry, Princeton University Press, Princeton, 1949

    MATH  Google Scholar 

  13. Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc., 117, 251–175 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  14. Shen, Y. B.: General solutions of some differential equations on Riemannian manifolds. J. of Math. Res. and Exp., 2, 55–60 (1982)

    Google Scholar 

  15. Bao, D., Robles, C.: Ricci and flag curvatures in Finsler geometry, In “A Sampler of Finsler Geometry” MSRI series, Cambridge University Press, 2004

  16. Bao, D., Shen, Z.: Finsler metrics of constant curvature on the Lie group S 3. J. of the London Math. Soc., 66, 453–467 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chen, X., Shen, Z.: Projectively flat Finsler metrics with almost isotropic S-curvature. Acta Math. Sci., Ser. B, 26, 307–313 (2006)

    Google Scholar 

  18. Mo, X., Shen, Z.: On negatively curved Finsler manifolds of scalar curvature. Canadian Math. Bull., 48, 112–120 (2005)

    MATH  MathSciNet  Google Scholar 

  19. Mo, X., Yang, C.: The explicit construction of Finsler metrics with special curvature properties. Diff. Geom. and its Appl.. Diff. Geom and its Appl., 24, 119–129 (2006)

    MATH  MathSciNet  Google Scholar 

  20. Shen, Z.: Lectures on Finsler geometry, World Scientific Publishers, Singapore, 2001

    MATH  Google Scholar 

  21. Shen, Z.: Landsberg Curvature, S-Curvature and Riemann Curvature, In “A Sampler of Finsler Geometry” MSRI series, Cambridge University Press, Cambridge, 2005

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, IUPUI, Indianapolis, IN, 46202-3216, USA

    Zhong Min Shen

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Hao Xing

Authors
  1. Zhong Min Shen
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  2. Hao Xing
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Correspondence to Zhong Min Shen.

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Supported by the National Natural Science Foundation of China (10371138)

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Shen, Z.M., Xing, H. On Randers metrics with isotropic S-curvature. Acta. Math. Sin.-English Ser. 24, 789–796 (2008). https://doi.org/10.1007/s10114-007-5194-0

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  • DOI: https://doi.org/10.1007/s10114-007-5194-0

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