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On Flag Curvature of Homogeneous Randers Spaces

Published online by Cambridge University Press:  20 November 2018

Shaoqiang Deng  and
Zhiguang Hu
Shaoqiang Deng
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People's Republic ofChina, e-mail: dengsq@nankai.edu.cn
Zhiguang Hu
Affiliation:
College of Mathematics, Tianjin Normal University, Tianjin 300387, People's Republic ofChina, e-mail: nankaitaiji@mail.nankai.edu.cn
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Abstract

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In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

Keywords

22E4653C30homogeneous Randers manifoldsflag curvatureDouglas spacestwo-step nilpotent Lie groups
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 1 , 01 February 2013 , pp. 66 - 81
Copyright
Copyright © Canadian Mathematical Society 2013

References

[AD08] An, H. and Deng, S., Invariant (α,β)-metrics on homogeneous manifolds. Monatsh. Math. 154(2008), no. 2, 89–102. http://dx.doi.org/10.1007/s00605-007-0529-1 Google Scholar
[BR04] Bao, D. and Robles, C., Ricci and flag curvatures in Finsler geometry. In: A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., 50, Cambridge University Press, Cambridge, 2004, pp. 197–259.Google Scholar
[BE87] Besse, A., Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987.Google Scholar
[BRV98] Berndt, J.,Ricci, F., and Vanhecke, L., Weakly symmetric groups of Heseiberg type. Differential Geom. Appl. 8(1998), no. 3, 275–284. http://dx.doi.org/10.1016/S0926-2245(98)00014-X Google Scholar
[BB76] Berard-Bergery, L., Les variétés Riemannienes homogénes simplement connexes de dimension impair à courbure strictement positive. J. Math. Pures Appl. 55(1976), no. 1, 47–68.Google Scholar
[BO49] Borel, A., Some remarks about Lie groups transitive on spheres and tori. Bull. Amer. Math. Soc. 55 (1949), 580–587. xhttp://dx.doi.org/10.1090/S0002-9904-1949-09251-0 Google Scholar
[CS05] Chern, S. S. and Shen, Z., Riemann-Finsler geometry. Nankai Tracts in Mathematics, 6,World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.Google Scholar
[DE08] Deng, S., The S-curvature of homogeneous Randers spaces. Differential Geom. Appl. 27(2009), no. 1, 75–84. http://dx.doi.org/10.1016/j.difgeo.2008.06.007 Google Scholar
[DH04] Deng, S., Invariant Randers metrics on homogeneous Riemannian manifold. J. Phys. A 37(2004), no. 15, 4353–4360; Corrigendum, ibid, 39(2006), 5249-5250. http://dx.doi.org/10.1088/0305-4470/37/15/004 Google Scholar
[DH07] Deng, S., Homogeneous Finsler spaces of negative curvature. J. Geometry Phys. 57(2007), no. 2, 657–664. http://dx.doi.org/10.1016/j.geomphys.2006.05.009 Google Scholar
[GO96]Gordon, C. S., Homogeneous Riemannian manifolds whose geodesics are orbits. In: Topics in geometry, Prog. Nonlinear Differential Equations Appl., 20, Birkhäuser Boston, Boston, MA, 1996, pp. 155–174.Google Scholar
[HE74]Heintze, E., On homogeneous manifolds of negative curvature. Math. Ann. 211(1974), 23–34. http://dx.doi.org/10.1007/BF01344139 Google Scholar
[HE78] Helgason, S., Differential geometry, Lie groups, and symmetric spaces. Pure and Applied Mathematics, 80, Academic Press, New York-London, 1978.Google Scholar
[HM07] Huang, L. and Mo, X., On curvature decreasing property of a class of navigation problems. Publ. Math. Debrecen 71(2007), no. 1-2, 141–163.Google Scholar
[HD11] Hu, Z. and Deng, S., Homogeneous Randers spaces with isotropic S-curvature and positive flag curvature. Math. Z., published online 2011, http://dx.doi.org/10.1007/s00209-010-0836-9. Google Scholar
[KA83]Kaplan, A., On the geometry of groups of Heisenberg type. Bull. London Math. Soc. 15(1983), no. 1, 35–42. http://dx.doi.org/10.1112/blms/15.1.35 Google Scholar
[KN63] Kobayashi, S. and Nomizu, K., Foundations of differential geometry. Interscience Publishers, New York-London, Vol. I, 1963; Vol. II, 1969.Google Scholar
[LA98] Lauret, J., Commutative spaces which are not weakly symmetric. Bull. London Math. Soc. 30(1998), no. 1, 29–36. http://dx.doi.org/10.1112/S0024609397003925 Google Scholar
[MS43] Montgomery, D. and Samelson, H., Transformation groups of spheres. Ann. Math. (2) 44(1943),454–470. http://dx.doi.org/10.2307/1968975 Google Scholar
[ON63] Oniščik, A. L., Transitive compact transformation groups. Math. Sb. (N.S.) 60(102)(1963), 447–485.Google Scholar
[SH03]Shen, Z., Finsler metrics with K = 0 and S = 0. Canad. J. Math. 55(2003), no. 1, 112–132. http://dx.doi.org/10.4153/CJM-2003-005-6 Google Scholar
[WA72] Wallach, N., Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. (2) 96(1972), 277–295. http://dx.doi.org/10.2307/1970789 Google Scholar
[WO64] A.Wolf, J., Homogeneity and bounded isometries in manifolds of negative curvature. Illinois J. Math. 8(1964), 14–18.Google Scholar
[WO07] A.Wolf, J., Harmonic analysis on commutative spaces. Mathematical Surveys and Monographs, 142, American Mathematical Society, Providence, RI, 2007.Google Scholar
[ZI96]Ziller, W., Weakly symmetric spaces. In: Topics in geometry, Progr. Nonlinear Differential Equations Appl., 20, Birkhäuser Boston, Boston, MA, 1996, pp. 355–368.Google Scholar