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Flag curvatures of the unit sphere in a Minkowski-Randers space | ||
| AUT Journal of Mathematics and Computing | ||
| دوره 2، شماره 2، آذر 2021، صفحه 275-282 اصل مقاله (315.15 K) | ||
| نوع مقاله: Original Article | ||
| شناسه دیجیتال (DOI): 10.22060/ajmc.2021.20237.1061 | ||
| نویسندگان | ||
| Libing Huang* ؛ Haibin Su | ||
| School of Mathematical Sciences, Nankai University, 94 Weijin Road, Tianjin 300071, P. R. China | ||
| چکیده | ||
| On a real vector space $V$, a Randers norm $\hat{F}$ is defined by $\hat{F}=\hat{\alpha}+\hat{\beta}$, where $\hat{\alpha}$ is a Euclidean norm and $\hat{\beta}$ is a covector. We show that the unit sphere $\Sigma$ in the Randers space $(V,\hat{F})$ has positive flag curvature, if and only if $|\hat{\beta}|_{\hat{\alpha}}< (5-\sqrt{17})/2 \approx 0.43845$, thus answering a problem proposed by Prof. Zhongmin Shen. Moreover, we prove that the flag curvature of $\Sigma$ has a universal lower bound $-4$. | ||
| کلیدواژهها | ||
| Flag curvature؛ Randers metric؛ Riemannian metric | ||
| مراجع | ||
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[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 [8] Z. Shen, Some open problems in Finsler geometry, unpublished note, 2009. | ||
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