理论数学  >> Vol. 5 No. 4 (July 2015)

局部射影平坦Berwald型(α, β)度量的一个刻画
A Characterization for Locally Projectively Flat Berwald Type (α, β)-Metrics

DOI: 10.12677/PM.2015.54023, PDF, HTML, XML, 下载: 1,835  浏览: 4,597  国家自然科学基金支持

作者: 余昌涛:华南师范大学数学科学学院,广东 广州

关键词: 芬斯勒几何 β)度量射影平坦Finsler Geometry β)-Metrics Projective Flatness

摘要: Berwald型(α, β)度量是形如F=(α, β)2/a的芬斯勒度量,其中α是一个黎曼度量,β是一个1形式。本文利用βαβ做一种特殊的度量形变,由此可以得到局部射影平坦Berwald型 (α, β)度量的一个刻画。该刻画不仅比其他研究者的相应方法和结论简单,而且从中我们可以看到局部射影平坦Berwald型(α, β)度量更为明确的几何结构。
Abstract: Berwald type (α, β)-metrics are those Finsler metrics expressed as F=(α, β)2/a, where α is a Riemannian metric, and β is a 1-form. In this paper, by using a special deformations for α and β due to β, we provide a characterization for locally projectively flat Berwald tpye (α, β)-metrics. Our characterization is simpler than the corresponding results of other researchers. Moreover, the geometrical structure of locally projectively flat Berwald type (α, β)-metrics is much clearer.

文章引用: 余昌涛. 局部射影平坦Berwald型(α, β)度量的一个刻画[J]. 理论数学, 2015, 5(4): 150-155. http://dx.doi.org/10.12677/PM.2015.54023

参考文献

[1] 沈一兵, 沈忠民 (2013) 现代芬斯勒几何初步. 高等教育出版社, 北京.
[2] Bao, D., Chern, S.-S. and Shen, Z.M. (2000) An introduction to riemann-finsler geometry. Graduate Texts in Mathe- matics, Volume 200, Springer.
http://dx.doi.org/10.1007/978-1-4612-1268-3
[3] Cheng, X.Y. and Shen, Z.M. (2012) Finsler geometry—An approach via randers spaces. Science Press, Beijing.
[4] Chern, S.-S. and Shen, Z.M. (2005) Riemann-finsler geometry. World Scientific, Singapore.
[5] Berwald, L. (1929) Über die n-dimensionalen Geometrien konstanter Krümmung. in denen die Geraden die kürzesten sind. Mathematische Zeitschrift, 30, 449-469.
http://dx.doi.org/10.1007/BF01187782
[6] Mo, X.H., Shen, Z.M. and Yang, C.H. (2006) Some constructions of projectively flat finsler metrics. Science in China (Series A), 49, 703-714.
http://dx.doi.org/10.1007/s11425-006-0703-7
[7] Li, B.L. and Shen, Z.M. (2007) On a class of projectively flat finsler metrics with constant flag curvature. International Journal of Mathematics, 18, 749.
http://dx.doi.org/10.1142/S0129167X07004291
[8] Zhou, L.F. (2010) A local classfication of a class of (α,β) metrics with constant flag curvature. Differential Geometry and its Applications, 28, 170-193.
http://dx.doi.org/10.1016/j.difgeo.2009.05.008
[9] Shen, Z.M. and Yu, C.T. (2014) On einstein square metrics. Publicationes Mathematicae-Debrecen, 85, 413-424.
http://dx.doi.org/10.5486/PMD.2014.6015
[10] Chen, B., Shen, Z.M. and Zhao, L.L. (2013) On a class of ricci-flat finsler metrics in finsler geometry. Journal of Geometry and Physics, 70, 30-38.
http://dx.doi.org/10.1016/j.geomphys.2013.03.009
[11] Sevim, E.S., Shen, Z.M. and Zhao, L.L. (2012) On a class of ricci-flat douglas metrics. International Journal of Mathematics, 23, 1250046.
http://dx.doi.org/10.1142/S0129167X12500462
[12] Shen, Z.M. and Yildirim, G.C. (2008) On a class of projectively flat metrics with constant flag curvature. Canadian Journal of Mathematics, 60, 443-456.
http://dx.doi.org/10.4153/CJM-2008-021-1