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On Some Non-Riemannian Quantities in Finsler Geometry

Published online by Cambridge University Press:  20 November 2018

Zhongmin Shen
Zhongmin Shen*
Affiliation:
Center of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang Province 310027, P.R. China and Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis (IUPUI), Indianapolis, IN 46202-3216, USA e-mail: [email protected]
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Abstract

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In this paper we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we study a new non-Riemannian quantity defined by the $\text{S}$-curvature. We show some relationships among the flag curvature, the $\text{S}$-curvature, and the new non-Riemannian quantity.

Keywords

53C6053B40Finsler metricS-curvaturenon-Riemannian quantity
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 1 , 01 March 2013 , pp. 184 - 193
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Akbar-Zadeh, H., Sur les espaces de Finsler à courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl. Sci. 74 (1988), no. 10, 271–322.Google Scholar
[2] Bao, D., Chern, and Shen, Z., An introduction to Riemann-Finsler geometry. Graduate Texts in Mathematics, 200, Springer-Verlag, New York, 2000.Google Scholar
[3] Bao, D., Robles, C., and Shen, Z., Zermelo navigation on Riemannian manifolds. J. Differential Geom. 66 (2004), no. 3, 377–435.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 (2003), no. 3, 762–780. http://dx.doi.org/10.1112/S0024610703004599 Google Scholar
[5] Chen, X. and Shen, Z., Randers metrics with special curvature properties. Osaka J. Math. 40 (2003), no. 1, 87–101.Google Scholar
[6] Chen, X. and Shen, Z., A class of Finsler metrics with isotropic S-curvature. Israel J. Math. 169 (2009), 317–340. http://dx.doi.org/10.1007/s11856-009-0013-1 Google Scholar
[7] Li, B. and Shen, Z., On Randers metrics of quadratic Riemann curvature. Internat. J. Math. 20 (2009), no. 3, 369–376. http://dx.doi.org/10.1142/S0129167X09005315 Google Scholar
[8] Mo, X., The flag curvature tensor on a closed Finsler space. Results Math. 36 (1999), no. 1–2, 149–159.Google Scholar
[9] Mo, X., On the flag curvature of a Finsler space with constant S-curvature. Houston J. Math. 31 (2005), no. 1, 131–144.Google Scholar
[10] Mo, X., On the non-Riemannian quantity H of a Finsler metric. Differential Geom. Appl. 27 (2009), no. 1, 7–14. http://dx.doi.org/10.1016/j.difgeo.2008.06.002 Google Scholar
[11] Mo, and Shen, On negatively curved Finsler manifolds of scalar curvature. Canad. Math. Bull. 48 (2005), no. 1, 112–120. http://dx.doi.org/10.4153/CMB-2005-010-3 Google Scholar
[12] 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. http://dx.doi.org/10.1007/s10711-007-9218-9 Google Scholar
[13] Shen, Z., Two-dimensional Finsler metrics with constant flag curvature. Manuscripta Math. 109 (2002), no. 3, 349–366. http://dx.doi.org/10.1007/s00229-002-0311-y Google Scholar
[14] Shen, Z. Differential geometry of spray and Finsler spaces. Kluwer Academic Publishers, Dordrecht, 2001.Google Scholar
[15] Shen, Z., Finsler manifolds with nonpositive flag curvature and constant S-curvature. Math. Z. 249 (2005), no. 3, 625–639. http://dx.doi.org/10.1007/s00209-004-0725-1 Google Scholar
[16] Shen, Z. and Yildirim, G. C., A characterization of Randers metrics of scalar flag curvature. http://www.math.iupui.edu/_zshen/Research/papers/ CharacterizationOfRandersMetricsOfScalarCurvature Shen.pdf Google Scholar