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A class of projectively flat and dually flat Finsler metrics

  • School of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China

Funds:  Supported by the National Natural Science Foundation of China (11371032).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.06.002
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  • Author Bio:

    HE Chao, male, born in 1992, master. Research field: Differential geometry. E-mail: begondjames@163.com

  • Corresponding author: SONG Weidong
  • Received Date: 10 March 2016
  • Rev Recd Date: 04 December 2016
  • Publish Date: 30 June 2017
    Abstract

  • Abstract

    Finsler geometry is just Riemannian geometry without quadratic restriction, and the projectively flat and dually flat Finsler metrics are very important in Finsler geometry. Here a class of 3-parameter of Finsler metrics were studied, and the necessary and sufficient conditions for the Finsler metrics to be projectively flat and dually flat were obtained.

    Abstract

    Finsler geometry is just Riemannian geometry without quadratic restriction, and the projectively flat and dually flat Finsler metrics are very important in Finsler geometry. Here a class of 3-parameter of Finsler metrics were studied, and the necessary and sufficient conditions for the Finsler metrics to be projectively flat and dually flat were obtained.
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    • References(13)
  • [1]
    SHEN B, ZHAO L. Some projectively at (α; β)-metrics[J]. Sci China Ser A: Math, 2006, 49: 838-851.
    [2]
    CHEN X, SHEN Z. Projectively at Finsler metrics with almost isotropic S-curvature[J]. Acta Mathmatica Scientia, 2006, 26: 307-313.
    [3]
    HILBERT D. Mathematical problems[J]. Bull Amer Math Soc, 2001, 37: 407-436.
    [4]
    MO X, YU C. On some explicit constructions of Finsler metrics with scalar flag curvature[J]. Canad J Math, 2010, 62: 1 325-1 339.
    [5]
    HUANG L, MO X. On spherically symmetric Finsler metrics of scalar curvature[J]. Journal of Geometry and Physics, 2012, 62: 2 279-2 287.
    [6]
    SONG W, ZHOU F. Spherically symmetric Finsler metrics with scalar flag curvature[J]. Turk J Math, 2015, 39(1):16-22.
    [7]
    HUANG L, MO X. Projectively at Finsler metrics with orthogonal invariance[J]. Annales Plonici Mathematical, 2013, 107: 259-270.
    [8]
    HUANG L, MO X. On some explicit constructions of dually at Finsler metrics[J]. Journal of Mathematical Analysis and Applications, 2013, 437: 675-683.
    [9]
    YU C, ZHU H. On a new class of Finsler metrics[J]. Differential Geometry and Its Applications, 2011, 29: 244-254.
    [10]
    HAMEL G. ber die Geometrieen in denen die Geraden die Kürzesten sind[J]. Math Ann, 1903, 57: 231-264.
    [11]
    SHEN Z. Riemann-Finsler geometry with applications to information geometry[J]. Chinese Annals of Mathematics, Series B, 2006, 27: 73-94.
    [12]
    HUANG L, MO X. On some dually at Finsler metrics with orthogonal invariance[J]. Nonlinear Analysis, 2014, 108: 214-222.
    [13]
    LI B. On dually at Finsler metrics[J]. Differential Geometry and Its Applications, 2013, 31: 718-721.
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    [1]
    SHEN B, ZHAO L. Some projectively at (α; β)-metrics[J]. Sci China Ser A: Math, 2006, 49: 838-851.
    [2]
    CHEN X, SHEN Z. Projectively at Finsler metrics with almost isotropic S-curvature[J]. Acta Mathmatica Scientia, 2006, 26: 307-313.
    [3]
    HILBERT D. Mathematical problems[J]. Bull Amer Math Soc, 2001, 37: 407-436.
    [4]
    MO X, YU C. On some explicit constructions of Finsler metrics with scalar flag curvature[J]. Canad J Math, 2010, 62: 1 325-1 339.
    [5]
    HUANG L, MO X. On spherically symmetric Finsler metrics of scalar curvature[J]. Journal of Geometry and Physics, 2012, 62: 2 279-2 287.
    [6]
    SONG W, ZHOU F. Spherically symmetric Finsler metrics with scalar flag curvature[J]. Turk J Math, 2015, 39(1):16-22.
    [7]
    HUANG L, MO X. Projectively at Finsler metrics with orthogonal invariance[J]. Annales Plonici Mathematical, 2013, 107: 259-270.
    [8]
    HUANG L, MO X. On some explicit constructions of dually at Finsler metrics[J]. Journal of Mathematical Analysis and Applications, 2013, 437: 675-683.
    [9]
    YU C, ZHU H. On a new class of Finsler metrics[J]. Differential Geometry and Its Applications, 2011, 29: 244-254.
    [10]
    HAMEL G. ber die Geometrieen in denen die Geraden die Kürzesten sind[J]. Math Ann, 1903, 57: 231-264.
    [11]
    SHEN Z. Riemann-Finsler geometry with applications to information geometry[J]. Chinese Annals of Mathematics, Series B, 2006, 27: 73-94.
    [12]
    HUANG L, MO X. On some dually at Finsler metrics with orthogonal invariance[J]. Nonlinear Analysis, 2014, 108: 214-222.
    [13]
    LI B. On dually at Finsler metrics[J]. Differential Geometry and Its Applications, 2013, 31: 718-721.
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