Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T19:32:43.806Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW QUANTITY IN RIEMANN-FINSLER GEOMETRY*

Published online by Cambridge University Press:  30 March 2012

XIAOHUAN MO ,
ZHONGMIN SHEN  and
HUAIFU LIU
XIAOHUAN MO
Affiliation:
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China e-mail: moxh@pku.edu.cn
ZHONGMIN SHEN
Affiliation:
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China e-mail: moxh@pku.edu.cn
HUAIFU LIU
Affiliation:
Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China e-mail: moxh@pku.edu.cn
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 note, we study a new Finslerian quantity Ĉ defined by the Riemannian curvature. We prove that the new Finslerian quantity is a non-Riemannian quantity for a Finsler manifold with dimension n = 3. Then we study Finsler metrics of scalar curvature. We find that the Ĉ-curvature is closely related to the flag curvature and the H-curvature. We show that Ĉ-curvature gives, a measure of the failure of a Finsler metric to be of weakly isotropic flag curvature. We also give a simple proof of the Najafi-Shen-Tayebi' theorem.

Keywords

58E20
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 3 , September 2012 , pp. 637 - 645
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Akbar-Zadeh, H., Sur les espaces de Finsler a courbures sectionnelles constants, Bull. Acad. Roy. Bel. Bull. Cl. Sci. 74 (5) (1988), 281–322.Google Scholar
2.Bai, Z., Shen, Y., Shui, N. and Guo, X., An introduction to Riemann geometry (Higher Education Press, 2004).Google Scholar
3.Chen, B. and Zhao, L., A note on Randers metrics of scalar flag curvature, Cana. Math. Bull. to appear.Google Scholar
4.Chen, X., Mo, X. and Shen, Z., On the flag curvature of Finsler metrics of scalar curvature, J. London Math. Soc. 68 (2) (2003), 762–780.CrossRefGoogle Scholar
5.Cheng, X. and Shen, Z., Randers metrics of scalar flag curvature, J. Aust. Math. Soc. 87 (2009), 359–370.CrossRefGoogle Scholar
6.Chern, S. S. and Shen, Z., Riemann-Finsler geometry, in Nankai tracts in mathematics, 6 (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005), x+192.CrossRefGoogle Scholar
7.Mo, X., On the non-Riemannian quantity H for a Finsler metric, Diff. Geom. Appl. 27 (2009), 7–14.CrossRefGoogle Scholar
8.Najafi, B., Shen, Z. and Tayebi, A., Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties, Geom. Dedicata 131 (2008), 87–97.CrossRefGoogle Scholar
9.Shen, Z., R-quadratic Finsler metrics, Publ. Math. Debrecen. 58 (2001), 263–274.CrossRefGoogle Scholar
10.Shen, Z., Differential geometry of spray and Finsler spaces (Kluwer Academic Publishers, 2001), 258.CrossRefGoogle Scholar
11.Shen, Z., On some non-Riemannian quantities in Finsler geometry, Cana. Math. Bull. (2011), to appear.Google Scholar
12.Shen, Z. and Xing, H., On Randers metrics with isotropic S-curvature, Acta Math. Sin. (Engl. Ser.) 24 (2008), 789–796.CrossRefGoogle Scholar
13.Tang, D., On the non-Riemannian quantity H in Finsler geometry, Diff. Geom. Appl. 29 (2011), 207–213.CrossRefGoogle Scholar