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On Projectively Flat (α, β)-metrics

Published online by Cambridge University Press:  20 November 2018

Zhongmin Shen
Zhongmin Shen*
Affiliation:
Center for Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang Province 310027, P.R. China e-mail: [email protected]
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Abstract

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The solutions to Hilbert's Fourth Problem in the regular case are projectively flat Finsler metrics. In this paper, we consider the so-called $\left( \alpha ,\,\beta \right)$-metrics defined by a Riemannian metric $\alpha$ and a 1-form $\beta$, and find a necessary and sufficient condition for such metrics to be projectively flat in dimension $n\,\ge \,3$.

Keywords

53B4053C60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 1 , 01 March 2009 , pp. 132 - 144
Copyright
Copyright © Canadian Mathematical Society 2009

References

[1] Chern, S.-S. and Shen, Z., Riemann-Finsler geometry. Nankai Tracts in Mathematics 6, World Scientific Publishing, Hackensack, NJ, 2005.Google Scholar
[2] Cheng, X. and Li, M., On a class of projectively flat (α, β)-metrics. Publ. Math. Debrecen 71(2007), no. 1–2, 195–205.Google Scholar
[3] Bácsó, S. and Matsumoto, M., On Finsler spaces of Douglas type–a generalization of the notion of Berwald space. Publ. Math. Debrecen 51(1997), no. 3–4, 385–406.Google Scholar
[4] Hamel, G., Über die Geometrieen in denen die Geraden die Kürzesten sind. Math. Ann. 57(1903), no. 2, 231–264.Google Scholar
[5] Li, B., Projectively flat Matsumoto metric and its approximation. ActaMath. Sci. Ser. B Engl. Ed. 27(2007), no. 4, 781–789.Google Scholar
[6] Mo, X., Shen, Z. and Yang, C., Some constructions of projectively flat Finsler metrics. Sci. China Ser. A, 49(2006), mp. 5, 703–714.Google Scholar
[7] Shen, Y. and Zhao, L., Some projectively flat (α, β)-metrics.. Sci. China Ser. A 49(2006), no. 6, 838–851.Google Scholar
[8] Shen, Z., Projectively flat Randers metrics with constant flag curvature. Math. Ann. 325(2003), no. 1, 19–30.Google Scholar
[9] Shen, Z. and Yildirim, G. C., On a class of projectively flat metrics with constant flag curvature. Canad. J. Math. 60(2008), no. 2, 443–456.Google Scholar
[10] Yu, Y., Projectively flat exponential Finsler metric. J. Zhejiang Univ. Science A, 7(2006), no. 6, 1068–1076.Google Scholar
[11] Yu, Y., Projectively flat arctangent Finsler metric. J. Zhejiang Univ. Science A, 7(2006), no. 12, 2097–2103.Google Scholar