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Journal of Zhejiang University SCIENCE A 2006 Vol.7 No.6 P.1068~1076

10.1631/jzus.2006.A1068


Projectively flat exponential Finsler metric


Author(s):  YU Yao-yong, YOU Ying

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310028, China

Corresponding email(s):   yuyaoyong@126.com

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


YU Yao-yong, YOU Ying. Projectively flat exponential Finsler metric[J]. Journal of Zhejiang University Science A, 2006, 7(6): 1068~1076.

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author="YU Yao-yong, YOU Ying",
journal="Journal of Zhejiang University Science A",
volume="7",
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year="2006",
publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2006.A1068"
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T1 - Projectively flat exponential Finsler metric
A1 - YU Yao-yong
A1 - YOU Ying
J0 - Journal of Zhejiang University Science A
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PB - Zhejiang University Press & Springer
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DOI - 10.1631/jzus.2006.A1068


Abstract: 
In this paper, we study a class of Finsler metric in the form F=αexp(&beta;/α)+εβ, where α is a Riemannian metric and &beta; is a 1-form, ε is a constant. We call F exponential Finsler metric. We proved that exponential Finsler metric F is locally projectively flat if and only if α is projectively flat and &beta; is parallel with respect to α. Moreover, we proved that the douglas tensor of exponential Finsler metric F vanishes if and only if &beta; is parallel with respect to α. And from this fact, we get that if exponential Finsler metric F is the Douglas metric, then F is not only a Berwald metric, but also a Landsberg metric.

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Reference

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[2] Bryant, R., 2002. Some remarks on Finsler manifolds with constant flag curvature. Houston J. Math, 28(2):221-262.

[3] Chen, X., Shen, Z., 2005. On Douglas metrics. Publ. Math. Debrecen, 66:503-512.

[4] Chern, S.S., Shen, Z., 2005. Riemann-Finsler Geometry. World Scientific, p.33.

[5] Hamel, G., 1993. Über die Geometrieen in denen die Geraden die kürzestensind. Math. Ann., 57(2):231-264.

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

[7] Mo, X., Shen, Z., Yang, C., 2006. Some constructions of projectively flat Finsler metric. Science in China—Series A: Mathematics, 49(5):703-714.

[8] Senarath, P., Thornley, G. M., 2004. Locally Projectively Flat Finsler Spaces with (α,β)-Metric. Http://www.natlib.govt.nz/files/bibliography/NZNB-1004.pdf.

[9] Shen, Z., 2003. Projectively flat Randers metrics of constant curvature. Math. Ann., 325(1):19-30.

[10] Shen, Z., 2004. Landsberg Curvature, S-curvature and Riemann Curvature, in a Sampler of Riemann-Finsler Geometry. MSRI Series Vol. 50. Cambridge University Press, p.303-355.

[11] Shen, Z., Civi Yildirim, G., 2005. On a class of projectively flat metrics of constant flag curvature. Canadian J. of Math., in press.

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