Flag curvatures of the unit sphere in a Minkowski-Randers space

Document Type : Original Article


1 School of Mathematical Sciences, Nankai University, P.R.China

2 School of Mathematical Sciences, Nankai University, P. R. China


On a real vector space V , a Randers norm Fˆ is defined by Fˆ = ˆα+βˆ, where ˆα is a Euclidean norm and βˆ is a covector. We show that the unit sphere Σ in the Randers space (V, Fˆ) has positive flag curvature, if and only if |βˆ|αˆ < (5 − √ 17)/2 ≈ 0.43845, thus answering a problem proposed by Prof. Zhongmin Shen. Moreover, we prove that the flag curvature of Σ has a universal lower bound −4.


Main Subjects

[1] D. Bao, S.-S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag, New York, 2000.
[2] D. Bao and C. Robles, Ricci and flag curvatures in Finsler geometry, pp.197-259 in A Sampler of Riemann-Finsler Geometry, edited by D. Bao et al., MSRI publ. 50, Cambridge Univ. Press, 2004.
[3] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66(3) (2004), 377-435. DOI 10.4310/jdg/1098137838.
[4] S.-S. Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices Amer. Math. Soc. 43(9) (1996), 959-963.
[5] P. Foulon, Geometrie des equations differentielles du second ordre, Annales de I’I.H.P.Physique theorique 45(1) (1986), 1-28.
[6] Z. Shen, Finsler metrics with K = 0 and S = 0, Canad. J. Math. 55(1) (2003), 112-132. DOI 10.4153/CJM[1]2003-005-6.
[7] Z. Shen and H. Xing, On Randers metrics of isotropic S-curvature, Acta Mathematica Sinica, English Series, 24 (2008), 789-796. [8] Z. Shen, Some open problems in Finsler geometry, unpublished note, 2009.