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Finsler Metrics with K = 0 and S = 0

Published online by Cambridge University Press:  20 November 2018

Zhongmin Shen
Zhongmin Shen*
Affiliation:
Math. Dept., IUPUI, 402 N. Blackford Street, Indianapolis, IN 46202-3216, USA, email: zshen@math.iupui.edu
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Abstract

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In the paper, we study the shortest time problem on a Riemannian space with an external force. We show that such problem can be converted to a shortest path problem on a Randers space. By choosing an appropriate external force on the Euclidean space, we obtain a non-trivial Randers metric of zero flag curvature. We also show that any positively complete Randers metric with zero flag curvature must be locally Minkowskian.

Keywords

53C6053B40
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 1 , 01 February 2003 , pp. 112 - 132
Copyright
Copyright © Canadian Mathematical Society 2003

References

[AIM] Antonelli, P. L., Ingarden, R. S. and Matsumoto, M., The theory of sprays and Finsler spaces with applications in physics and biology. Fund. Theories Phys. 58, Kluwer Academic Publishers, 1993.Google Scholar
[AZ] Akbar-Zadeh, H., Sur les espaces de Finsler à courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), 281–322.Google Scholar
[BaChSh] Bao, D., Chern, S. S. and Shen, Z., An introduction to Riemann-Finsler geometry. Springer, 2000.Google Scholar
[BaMa] Bácsó, S. and Matsumoto, M., On Finsler spaces of Douglas type. A generalization of the notion of Berwald space. Publ. Math. Debrecen 51 (1997), 385–406.Google Scholar
[BaRo] Bao, D. and Robles, C., On Randers metrics of constant curvature. Rep.Math. Phys., to appear.Google Scholar
[BaSh] Bao, D. and Shen, Z., Finsler metrics of constant curvature on the Lie group S3. J. London Math. Soc., to appear.Google Scholar
[Be] Berwald, L., Über eine characteristic Eigenschaft der allgemeinen Raüme konstanter Krümmung mit gradlinigen Extremalen. Monatsh.Math. Phys. 36 (1929), 315–330.Google Scholar
[Br1] Bryant, R., Finsler structures on the 2-sphere satisfying K = 1. In: Finsler Geometry, Contemp.Math. 196, Amer. Math. Soc., Providence, RI, 1996, 27–42.Google Scholar
[Br2] Bryant, R., Projectively flat Finsler 2-spheres of constant curvature. Selecta Math. (N.S.) 3 (1997), 161–203.Google Scholar
[Br3] Bryant, R., Finsler manifolds with constant curvature. Talk at the 1998 Geometry Festival in Stony Brook.Google Scholar
[Ma] Matsumoto, M., Randers spaces of constant curvature. Rep. Math. Phys. 28 (1989), 249–261.Google Scholar
[Ra] Randers, G., On an asymmetric metric in the four-space of general relativity. Phys. Rev. 59 (1941), 195–199.Google Scholar
[Sh1] Shen, Z., Differential Geometry of Spray and Finsler Spaces. Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar
[Sh2] Shen, Z., Funk metrics and R-flat sprays. Preprint, unpublished.Google Scholar
[Sh3] Shen, Z., On R-quadratic Finsler spaces. Publ. Math. Debrecen 58 (2001), 263–274.Google Scholar
[Sh4] Shen, Z., Volume comparison and its applications in Riemann-Finsler geometry. Adv. Math. 128 (1997), 306–328.Google Scholar
[Sh5] Shen, Z., Two-dimensional Finsler metrics with constant curvature.Manuscripta Math., to appear.Google Scholar
[Sh6] Shen, Z., Projectively flat Randers metrics with constant curvature.Math. Ann., to appear.Google Scholar
[Sh7] Shen, Z., Lectures on Finsler Geometry. World Scientific, Singapore, 2001.Google Scholar
[SSAY] Shibata, C., Shimada, H., Azuma, M. and Yasuda, H., On Finsler spaces with Randers’ metric. Tensor (N.S.) 31 (1977), 219–226.Google Scholar
[Sz] Szabó, Z. I., Positive definite Berwald spaces (structure theorems on Berwald spaces). Tensor (N.S.) 35 (1981), 25–39.Google Scholar
[YaSh] Yasuda, H. and Shimada, H., On Randers spaces of scalar curvature. Rep. Math. Phys. 11 (1977), 347–360.Google Scholar