Rank inequality in homogeneous Finsler geometry

Document Type : Review Article


School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.R. China


This is a survey on some recent progress in homogeneous Finsler geometry. Three topics are discussed, the classification of positively curved homogeneous Finsler spaces, the geometric and topological properties of homogeneous Finsler spaces satisfying $K\geq0$ and the (FP) condition, and the orbit number of prime closed geodesics in a compact homogeneous Finsler manifold. These topics share the same similarity that the same rank inequality, i.e., $\mathrm{rank}G\leq\mathrm{rank}H+1$ for $G/H$ with compact $G$ and $H$, plays an important role. In this survey, we discuss in each topic how the rank inequality is proved, explain its importance, and summarize some relevant results.


Main Subjects

[1] D. Anosov, Geodesics in Finsler geometry (in Russian), in: Proc. I.C.M., Montreal, Canad. Math. Congress 2 (1975), 293-297. English translation in A.M.S. Transl. 109 (1977), 81-85.
[2] J. C. Alvarez Paive and C. E. Duran, Isometric submersion of Finsler manifolds, Proc. Amer. Math. Soc. 129 (2001), 2409-2417.
[3] S. Aloff and N. Wallach, An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93-97.
[4] A. Arvanitoyeorgos and Y. Wang, Homogeneous geodesics in generalized Wallach spaces, Bull. Belgian Math. Soc. - Simon Stevin 24(2) (2016), 257-270.
[5] Y. Bazaikin, On a certain class of 13-dimensional Riemannian manifolds with positive curvature, Sib. Math. J. 37(6) (1996), 1219-1237.
[6] M. Berger, Les vari´et´es riemanniennes homog`enes normales simplement connexes `a courbure strictement positive, Ann. Scuola Norm. Sup. Pisa (3) 15 (1961), 179-246.
[7] M. Berger, Trois remarques sur les vari´et´es riemanniennes `a courbure positive, C. R. Acad. Sci. Paris, Ser A-B, 263 (1966), 76-78.
[8] L. B´erard Bergery, Les vari´etes Riemannienes homog´enes simplement connexes de dimension impair `a courbure strictement positive, J. Math. Pure Appl. 55 (1976), 47-68.
[9] A. Borel, Some remarks about Lie groups transitive on spheres and tori, Bull. Amer. Math. Soc. 55 (1940), 580-587.
[10] G. E. Bredon, Introduction ot compact transformation groups, Pure and Applied Mathematics, Vol 46, Aca[1]demic Press, New York, 1972.
[11] D. Bao, S. S. Chern and Z. Shen, An introduction to Riemann-Finsler geometry, Springer-Verlag, New York, 2000.
[12] R. Bryant, P. Foulon, S. Ivanov, V. Matveev and W. Ziller, Geodesic behavior for Finsler metrics of constant positive flag curvature on S 2 , J. Diff. Geom. 117(1) (2021), 1-22.
[13] V. Bangert and Y. Long, The existence of two closed geodesics on every Finsler manifolds, Math. Ann. 346 (2010), 335-366.
[14] V. Berestovskii and Yu. G. Nikonorov, On δ-homogeneous Riemannian manifolds, Diff. Geom. Appl. 26 (2008), 514-535.
[15] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differ. Geom. 66(3) (2004), 377-435.
[16] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North Holland - American Elsevier, 1975.
[17] S. Deng, Fixed points of isometries of a Finsler space, Publ. Math. Debrecen, 72 (2008), 469-474.
[18] O. Dearricott, A 7-dimensional manifold with positive curvature, Duke Math. J. 158 (2011), 307-346.
[19] S. Deng, Homogeneous Finsler Spaces, Springer, New York, 2012.
[20] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific J. Math. 207(1) (2002), 149-155.
[21] S. Deng and Z. Hu, Curvatures of homogeneous Randers spaces, Adv. Math. 240 (2013), 194-226.
[22] H. Duan, Y. Long and W. Wang, Two closed geodesics on compact simply connected bumpy Finsler manifolds, J. Diff. Geom. 104 (2016), 275-289.
[23] S. Deng and M. Xu, Clifford-Wolf translations of Finsler spaces, Forum Math. 26 (2014), 1413-1428.
[24] S. Dengt and M. Xu, Recent progress on homogeneous Finsler spaces with positive curvature, European J. Math. 3(4) (2017) S.I. 974-999.
[25] J. Eschenburg, New examples of manifolds of positive curvature, Invent. Math. 66 (1982), 469-480.
[26] J. Eschenburg, Inhomogeneous spaces of positive curvature, Diff. Geom. Appl. 2 (1992), 123-132.
[27] P. Foulon, Ziller-Katok deformations of Finsler metrics, in:2004 International Symposium on Finsler geometry, Tianjin, 2004, 22-24.
[28] A. Fet, A periodic problem in the calculus of variations (in Russian), Dokl. Akad Nauk SSSR 160 (1965), 287-289; English translation in Soviet Math. 6 (1965), 85-88.
[29] T. Frankel, Manifolds with positive curvature, Pac. J. Math. 11(1) (1961), 165-174.
[30] P. Foulon and V. S. Matveev, Zermelo deformation of Finsler metrics by Killing vector fields, Electron. Res. Announc. Math. Sci. 25 (2018), 1-7.
[31] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer-Verlag, 1987.
[32] K. Grove, L. Verdiani and W. Ziller, An exotic T1S 4 with positive curvature, Geom. Funct. Anal. 21 (2011), 499-521.
[33] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, San Diego, 1978.
[34] L. Huang, On the fundamental equations of homogeneous Finsler spaces, Differential Geom. Appl. 40 (2015), 187-208.
[35] L. Huang, Flag curvatures of homogeneous Finsler spaces, European J. Math. 3 (2017), 1000-1029.
[36] Z. Hu and S. Deng, Homogeneous Randers spaces with positive flag curvature and isotropic S-curvature, Math. Z. 270 (2012), 989-1009.
[37] L. Huang, X. Mo, On curvature decreasing property of a class of navigation problems, Publ. Math. Debrecen, 71 (2007), 991-996.
[38] A. Katok, Ergodic properties of degenerate integrable Hamiltonian systems, Izv Akad Nauk SSSR, 37 (1973), 535-571.
[39] W. Klingenberg, Lectures on Closed Geodesics, Springer, Berlin, 1978.
[40] O. Kowalski and L. Vanhecke, Riemannian manifolds with homogeneous geodesics, Boll. Unione Mat. Ital. B (7) 5(1) (1991), 189-246.
[41] M. Matsumoto, Theory of Finsler spaces with (α, β)-metrics, Rep. Math. Phys. 31 (1992), 43-83.
[42] S. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401-404.
[43] D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. Math. 44 (1943), 454-470.
[44] Yu.G. Nikonorov, On the structure of geodesic orbit Riemannian spaces, Ann. Glob. Anal. Geom. 52(3) (2017), 289-311.
[45] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59 (1941), 195-199.
[46] H. Rademacher, The second geodesic on Finsler spheres of dimension n > 2, Trans. Amer. Math. Soc. 362 (2010), 1413-1421.
[47] Z. Shen, Volume comparison and its applications in Riemann-Finsler geometry, Adv. Math. vol. 128 (1997), 306-328.
[48] Z. Shen, Lectures on Finsler geometry, World Scientific, 2001.
[49] J. Synge, The first and second variations of length in Riemannian space, Proc. London Math. Soc. 25 (1926), 247-264.
[50] J. Synge, On the connectivity of spaces of positive curvature, Quart. J. Math. 7 (1936), 316-320.
[51] L. Verdiani and W. Ziller, Positively curved homogeneous metrics on spheres, Math. Z. 261 (2009), 473-488.
[52] N. R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 (1972), 277-295.
[53] W. Wang, On a conjecture of Anosov, Adv. Math. 230 (2012), 1597-1617.
[54] B. Wilking, The normal homogeneous space SU(3)×SU(3)/U∗ (2) has positive sectional curvature, Proc. Amer. Math. Soc. 127 (1999), 1191-1194.
[55] B. Wilking and W. Ziller, Revisiting homogeneous spaces with positive curvature, Journal f¨ur die reine und angewandte Mathematik, 738 (2018), 313-328.
[56] M. Xu, Examples of flag-wise positively curved spaces, Diff. Geom. Appl. 52 (2017), 42-50.
[57] M. Xu, Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics, Pac. J. Math. 302 (2019), 353-370.
[58] M. Xu, Homogeneous Finsler spaces with only one orbit of prime closed geodesics, Sci. China Math. 63(11) (2020), 2321-2342.
[59] M. Xu, Geodesic orbit Finsler space with K ≥ 0 and the (FP) condition, Adv. Geom. (2021), to appear.
[60] M. Xu and S. Deng, Killing frames and S-curvature of homogeneous Finsler spaces, Glasgow Math. J. 567 (2015), 457-464.
[61] M. Xu and S. Deng, Homogeneous (α, β)-spaces with positive flag curvature and vanishing S-curvature, Non[1]linear Ana. 127 (2015), 45-54.
[62] M. Xu and S. Deng, Normal homogeneous Finsler spaces, Transform. Groups. 22(4) (2017), 1143-1183.
[63] M. Xu and S. Deng, Towards the classification of odd dimensional homogeneous reversible Finsler spaces with positive flag curvature, to appear in Ann. Mat. Pura Appl. (4) 196(4) (2017), 1459-1488.
[64] M. Xu and S. Deng, Homogeneous Finsler spaces and the flag-wise positively curved condition, Forum Math. 30 (6) (2018), 1521-1537.
[65] M. Xu and S. Deng, L. Huang and Z. Hu, Even dimensional homogeneous Finsler spaces with positive flag curvature, Indiana Univ. Math. J. 66(3) (2017), 949-972.
[66] M. Xu and J. A. Wolf, Sp(2)/U(1) and a positive curvature problem, Differential Geom. Appl. 42 (2015), 115-124. [67] M. Xu and W. Ziller, Reversible homogeneous Finsler metrics with positive flag curvature, Forum Math. 29(5) (2017), 1212-1226.
[68] M. Xu and L. Zhang, δ-homogeneity in Finsler geometry and the positive curvature problem, Osaka J. Math. 55(1) (2018), 177-194.
[69] Z. Yan and S. Deng, Finsler spaces whose geodesics are orbits, Diff. Geom. Appl. 36 (2014), 1-23.
[70] W. Ziller, Geometry of the Katok examples, Ergodic Theory Dynam. Systems, 3 (1982), 135-137.
[71] W. Ziller, Riemannian manifolds with positive sectional curvature, in: Geometry of manifolds with non-negative sectional curvature, ed. R. Herrera and L. Hernandez-Lamoneda, Lecture Notes in Mathematics 2110, 2014.
[72] L. Zhang and M. Xu, Standard homogeneous (α1, α2)-metrics and geodesic orbit property, preprint (2019), arXiv:1912.00210.