Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T05:41:34.312Z Has data issue: false hasContentIssue false
  • English
  • Français

On Negatively Curved Finsler Manifolds of Scalar Curvature

Published online by Cambridge University Press:  20 November 2018

Xiaohuan Mo  and
Zhongmin Shen
Xiaohuan Mo
Affiliation:
LMAM, School of Mathematical Sciences, Beijing University, Beijing 100871, P.R. China e-mail: moxh@pku.edu.cn
Zhongmin Shen
Affiliation:
Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, IN 46202-3216, U.S.A. e-mail: zshen@math.iupui.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension $n\,\ge \,3$. We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type. We also study the case when the Finsler metric is locally projectively flat.

Keywords

53C60
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 1 , 01 March 2005 , pp. 112 - 120
Copyright
Copyright © Canadian Mathematical Society 2005

References

[AZ] Akbar-Zadeh, H., Sur les espaces de Finsler á courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl, Sci. (5)94(1988), 281–322.Google Scholar
[BCS] Bao, D., Chern, S. S. and Shen, Z., An Introduction to Riemann-Finsler Geometry. Springer-Verlag, New York, 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 spaces of constant flag curvature. Rep.Math. Phys. 51(2003), 9–42.Google Scholar
[BaRoSh] Bao, D., Robles, C. and Shen, Z., Zermelo navigation on Riemannian manifolds. J. Diff. Geom., to appear.Google Scholar
[Be1] Berwald, L., Untersuchung der Krümmung allgemeiner metrischer Räume auf Grund des in ihnen herrschenden Parallelismus. Math. Z. 25(1926), 40–73.Google Scholar
[Be2] Berwald, L., Parallelübertragung in allgemeinen Räumen. Atti Congr. Intern.Mat. Bologn. 4(1928), 263–270.Google Scholar
[Be3] Berwald, L., über eine characteristic Eigenschaft der allgemeinen Räume konstanter Krümmung mit gradlinigen Extremalen. Monatsh. Math. Phys. 36(1929), 315–330.Google Scholar
[Be4] Berwald, L., über die n-dimensionalen Geometrien konstanter Krümmung, in denen die Geraden die kürzesten sind. Math. Z. 30(1929), 449–469.Google Scholar
[ChSh] Chen, X. and Shen, Z., Randers metrics with special curvature properties. Osaka J. Math. 40(2003), 87–102.Google Scholar
[Ch] Chern, S. S., On the Euclidean connections in a Finsler space. Proc. National Acad. Sci. 29(1943), 33–37.Google Scholar
[Fk] Funk, P., Über Geometrien bei denen die Geraden die Kürzesten sind. Math. Ann. 101(1929), 226–237.Google Scholar
[Ma1] Matsumoto, M., On C-reducible Finsler spaces. Tensor, N.S. 24(1972), 29–37.Google Scholar
[Ma2] Matsumoto, M., Randers spaces of constant curvature. Rep. Math. Phys. 28(1989), 249–261.Google Scholar
[MaHo] Matsumoto, M. and Hōjō, S., A conclusive theorem for C-reducible Finsler spaces. Tensor. N. S. 32(1978), 225–230.Google Scholar
[MaSh] Matsumoto, M. and Shimada, H., The corrected fundamental theorem on the Randers spaces of constant curvature. Tensor, N.S. (to appear).Google Scholar
[Mo1] Mo, X., The flag curvature tensor on a closed Finsler space. Results Math. 36(1999), 149–159.Google Scholar
[Mo2] Mo, X., On the flag curvature of a Finsler space with constant S-curvature. preprint.Google Scholar
[Nu] Numata, S., On Landsberg spaces of scalar curvature. J. Korea Math. Soc. 12(1975), 97–100.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., On projectively related Einstein metrics in Riemann-Finsler geometry. Math. Ann. 320(2001), 625–647.Google Scholar
[Sh2] Shen, Z., Finsler metrics with K= 0 and S= 0 . Canadian J. Math. 55(2003), 112–132.Google Scholar
[Sh3] Shen, Z., Two-dimensional Finsler metrics with constant flag curvature. Manuscripta Math. 109(2002), 349–366.Google Scholar
[Sh4] Shen, Z., Lectures on Finsler Geometry. World Scientific Publishing, Singapore, 2001.Google Scholar
[Sh5] Shen, Z., Differential Geometry of Spray and Finsler Spaces. Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar
[YaSh] Yasuda, H. and Shimada, H., On Randers spaces of scalar curvature. Rep. Math. Phys. 11(1977), 347–360.Google Scholar