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An equivalence theorem of a class of Minkowski norms and its applications

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Abstract

In this paper, the Cartan tensors of the (α, β)-norms are investigated in detail. Then an equivalence theorem of (α, β)-norms is proved. As a consequence in Finsler geometry, general (α, β)-metrics on smooth manifolds of dimension n ⩾ 4 with vanishing Landsberg curvatures must be Berwald manifolds.

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References

  1. Aikou T. Some remarks on Berwald manifolds and Landsberg manifolds. Acta Math Acad Paedagog Nyházi (NS), 2010, 26: 139–148

    MathSciNet  MATH  Google Scholar 

  2. Antonelli P L, Ingarden R S, Matsumoto M. The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Dordrecht: Springer, 1993

    Book  Google Scholar 

  3. Asanov G S. Finsleroid-Finsler spaces of positive-definite and relativistic types. Rep Math Phys, 2006, 58: 275–300

    Article  MathSciNet  Google Scholar 

  4. Asanov G S. Finsleroid-Finsler space and geodesic spray coefficients. Publ Math Debrecen, 2007, 71: 397–412

    MathSciNet  MATH  Google Scholar 

  5. Bácsó S, Ilosvay F, Kiss B. Landsberg spaces with common geodesics. Publ Math Debrecen, 1993, 42: 139–144

    MathSciNet  MATH  Google Scholar 

  6. Bácsó S, Matsumoto M. Reduction theorems of certain Landsberg spaces to Berwald spaces. Publ Math Debrecen, 1996, 48: 357–366

    MathSciNet  MATH  Google Scholar 

  7. Bao D. On two curvature-driven problems in Riemann-Finsler geometry. In: Advanced Studies in Pure Mathematics, vol. 48. Tokyo: Math Soc Japan, 2007, 19–71

    MATH  Google Scholar 

  8. Bao D, Chern S-S, Shen Z. An Introduction to Riemann-Finsler Geometry. Graduate Texts in Mathematics, vol. 200. New York: Springer-Verlag, 2000

    Book  Google Scholar 

  9. Castro I. Lagrangian spheres in the complex Euclidean space satisfying a geometric equality. Geom Dedicata, 1998, 70: 197–208

    Article  MathSciNet  Google Scholar 

  10. Chern S-S, Shen Z. Riemann-Finsler Geometry. Nankai Tracts in Mathematics, vol. 6. Hackensack: World Scientific, 2005

    Book  Google Scholar 

  11. Crampin M. On Landsberg spaces and the Landsberg-Berwald problem. Houston J Math, 2011, 37: 1103–1124

    MathSciNet  MATH  Google Scholar 

  12. Crampin M. A condition for a Landsberg space to be Berwaldian. Publ Math Debrecen, 2018, 93: 143–155

    Article  MathSciNet  Google Scholar 

  13. Feng H, Li M. Adiabatic limit and connections in Finsler geometry. Comm Anal Geom, 2013, 21: 607–624

    Article  MathSciNet  Google Scholar 

  14. Ichijyo Y. On special Finsler connections with the vanishing hv-curvature tensor. Tensor (NS), 1978, 32: 149–155

    MathSciNet  MATH  Google Scholar 

  15. Li A, Simon U, Zhao G. Global Affine Differential Geometry of Hypersurfaces. Berlin-New York: Walter de Gruyter, 1993

    Book  Google Scholar 

  16. Li M. Equivalence theorems of Minkowski spaces and applications in Finsler geometry (in Chinese). Acta Math Sinica (Chin Ser), 2019, 62: 177–190

    MathSciNet  MATH  Google Scholar 

  17. Li M, Zhang L. Properties of Berwald scalar curvature. Front Math China, 2020, 15: 1143–1153

    Article  MathSciNet  Google Scholar 

  18. Matsumoto M. On C-reducible Finsler spaces. Tensor (NS), 1972, 24: 29–37

    MathSciNet  MATH  Google Scholar 

  19. Matsumoto M. Remarks on Berwald and Landsberg spaces. Contemp Math, 1996, 196: 79–82

    Article  MathSciNet  Google Scholar 

  20. Matsumoto M, Shibata C. On semi-C-reducibility, T-tensor = 0 and S4-likeness of Finsler spaces. J Math Kyoto Univ, 1979, 19: 301–314

    MathSciNet  MATH  Google Scholar 

  21. Matveev V S. On “All regular Landsberg metrics are always Berwald” by Z I Szabó. Balkan J Geom Appl, 2009, 14: 50–52

    MathSciNet  MATH  Google Scholar 

  22. Muzsnay Z. The Euler-Lagrange PDE and Finsler metrizability. Houston J Math, 2006, 32: 79–98

    MathSciNet  MATH  Google Scholar 

  23. Nomizu K, Sasaki T. Affine Differential Geometry. Cambridge: Cambridge University Press, 1994

    MATH  Google Scholar 

  24. Schneider R. Zur affinen differentialgeometrie im Großen. I. Math Z, 1967, 101: 375–406

    Article  MathSciNet  Google Scholar 

  25. Shen Z. On a class of Landsberg metrics in Finsler geometry. Canad J Math, 2009, 61: 1357–1374

    Article  MathSciNet  Google Scholar 

  26. Shima H. The Geometry of Hessian Structure. Singapore: World Scientific, 2007

    Book  Google Scholar 

  27. Simon U, Schwenk-Schellschmidt A, Viesel H. Introduction to the Affine Differential Geometry of Hypersurfaces. Tokyo: Science University of Tokyo, 1991

    MATH  Google Scholar 

  28. Song P, Li M. On some properties of the Cartan type one form of a Finsler manifold. J Southwest Univ (Nat Sci Ed), 2014, 36: 119–123

    Google Scholar 

  29. Szabó Z I. All regular Landsberg metrics are Berwald. Ann Global Anal Geom, 2008, 34: 381–386

    Article  MathSciNet  Google Scholar 

  30. Szabó Z I. Correction to “All regular Landsberg metrics are Berwald”. Ann Global Anal Geom, 2009, 35: 227–230

    Article  MathSciNet  Google Scholar 

  31. Torromé R G. On the Berwald-Landsberg problem. arXiv:1110.5680v4, 2011

  32. Yu C, Zhu H. On a new class of Finsler metrics. Differential Geom Appl, 2011, 29: 244–254

    Article  MathSciNet  Google Scholar 

  33. Zhang W. Lectures on Chern-Weil Theory and Witten Deformations. Nankai Tracts in Mathematics, vol. 4. Hackensack: World Scientific, 2001

    Book  Google Scholar 

  34. Zhou S, Li B. On Landsberg general (α, β)-metrics with a conformal 1-form. Differential Geom Appl, 2018, 59: 46–65

    Article  MathSciNet  Google Scholar 

  35. Zhou S S, Wang J Y, Li B L. On a class of almost regular Landsberg metrics. Sci China Math, 2019, 62: 935–960

    Article  MathSciNet  Google Scholar 

  36. Zohrehvand M, Maleki H. On general (α, β)-metrics of Landsberg type. Int J Geom Methods Mod Phys, 2016, 13: 1650085

    Article  MathSciNet  Google Scholar 

  37. Zou Y, Cheng X. The generalized unicorn problem on (α, β)-metrics. J Math Anal Appl, 2014, 414: 574–589

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11221091, 11271062, 11501067, 11571184, 11871126 and 11931007), China Scholarship Council Visiting Scholar Program and the Fundamental Research Funds for the General Universities and Nankai Zhide Foundation. The third author expresses his appreciation to Professor Guofang Wang for his warm hospitality and discussions on mathematics, while he stayed in Mathematisches Institut of Albert-Ludwigs-Universität Freiburg in 2018.

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Authors and Affiliations

  1. Chern Institute of Mathematics & LPMC, Nankai University, Tianjin, 300071, China

    Huitao Feng & Yuhua Han

  2. Mathematical Science Research Center, Chongqing University of Technology, Chongqing, 400054, China

    Ming Li

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  1. Huitao Feng
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  2. Yuhua Han
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  3. Ming Li
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Correspondence to Ming Li.

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In Memory of Professor Zhengguo Bai (1916–2015)

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Feng, H., Han, Y. & Li, M. An equivalence theorem of a class of Minkowski norms and its applications. Sci. China Math. 64, 1429–1446 (2021). https://doi.org/10.1007/s11425-020-1812-3

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  • DOI: https://doi.org/10.1007/s11425-020-1812-3

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