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Journal of Zhejiang University SCIENCE A 2007 Vol.8 No.6 P.963-968

http://doi.org/10.1631/jzus.2007.A0963


Projectively flat Asanov Finsler metric


Author(s):  HAN Jing-wei, YU Yao-yong

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China; more

Corresponding email(s):   jingweih@126.com

Key Words:  Exponential Finsler metric, Projectively flat, (&alpha, , &beta, )-metrics, Douglas tensor


HAN Jing-wei, YU Yao-yong. Projectively flat Asanov Finsler metric[J]. Journal of Zhejiang University Science A, 2007, 8(6): 963-968.

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author="HAN Jing-wei, YU Yao-yong",
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T1 - Projectively flat Asanov Finsler metric
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Abstract: 
In this work, we study the Asanov Finsler metric F=α(&beta;2/α2+/α+1)1/2exp{(G/2)arctan[&beta;/()+G/2]}, where α=(&alpha;ijyiyj)1/2 is a Riemannian metric and &beta;=biyj is a 1-form, g∈(−2,2), h=(1−g2/4)1/2, G=g/h. We give the necessary and sufficient condition for Asanov metric to be locally projectively flat, i.e., α is projectively flat and &beta; is parallel with respect to α. Moreover, we proved that the douglas tensor of Asanov Finsler metric vanishes if and only if &beta; is parallel with respect to α.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

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[3] Chern, S.S., Shen, Z., 2005. Riemann-Finsler Geometry. World Scientific, Singapore, p.53.

[4] Hamel, G., 1903. Über die geometrieen in denen die Geraden die Kürzesten sind. Math. Ann., 57:231-264.

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[6] Matsumoto, M., 1998. Finler spaces with (α,β)-metric of Douglas type. Tensor, N. S., 60:123-134.

[7] Senarath, P., Thornley, G.M., 2004. Locally projectively flat Finsler spaces with (α,β)-metrics. Manuscript.

[8] Shen, Z., 2003. Projectively flat Randers metrics of constant curvature. Math. Ann., 325:19-30.

[9] Shen, Z., 2004. Landsberg Curvature, S-curvature and Riemann Curvature, in a Sampler of Riemann-Finsler Geometry. MSRI Series, Vol. 50. Cambridge University Press.

[10] Shen, Z., Civi Yildirim, G., 2005. On a class of projectively flat metrics of constant flag curvature. Can. J. Math., in press. Http://www.math.iupui.edu/~zshen/Research/preprintindex.html

[11] Shen, Z., 2006. On projectively flat (α,β)-metrics. Http://www.math.iupui.edu/~zshen/Research/preprintindex.html

[12] Yu, Y.Y., You, Y., 2006. Projectively flat exponential Finsler metric. J. Zhejiang Univ. Sci. A, 7(6):1068-1076.

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