Abstract
Studying geodesic orbit Randers metrics on spheres, we obtain a complete classification of such metrics. Our method relies upon the classification of geodesic orbit Riemannian metrics on the spheres Sn in [17] and the navigation data in Finsler geometry. We also construct some explicit U(n + 1)-invariant metrics on S2n+1 and Sp(n + 1)U(1)-invariant metrics on S4n+3.
Communicated by: J. Bamberg
Funding: The second author is supported by NSFC (Nos. 11626134, 11701300) and the K. C. Wong Magna Fund in Ningbo University.
Acknowledgements
We are deeply grateful to the referee for a very careful reading of the manuscript and helpful suggestions.
References
[1] D. Alekseevsky, A. Arvanitoyeorgos, Riemannian flag manifolds with homogeneous geodesics. Trans. Amer. Math. Soc. 359 (2007), 3769–3789. MR2302514 Zbl 1148.5303810.1090/S0002-9947-07-04277-8Search in Google Scholar
[2] D. V. Alekseevsky, Y. G. Nikonorov, Compact Riemannian manifolds with homogeneous geodesics. SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 093, 16 pages. MR2559668 Zbl 1189.5304710.3842/SIGMA.2009.093Search in Google Scholar
[3] D. Bao, S.-S. Chern, Z. Shen, An introduction to Riemann-Finsler geometry. Springer 2000. MR1747675 Zbl 0954.5300110.1007/978-1-4612-1268-3Search in Google Scholar
[4] D. Bao, C. Robles, Ricci and flag curvatures in Finsler geometry. In: A sampler of Riemann-Finsler geometry, volume 50 of Math. Sci. Res. Inst. Publ., 197–259, Cambridge Univ. Press 2004. MR2132660 Zbl 1076.53093Search in Google Scholar
[5] A. Borel, Some remarks about Lie groups transitive on spheres and tori. Bull. Amer. Math. Soc. 55 (1949), 580–587. MR29915 Zbl 0034.0160310.1090/S0002-9904-1949-09251-0Search in Google Scholar
[6] S. Deng, The S-curvature of homogeneous Randers spaces. Differential Geom. Appl. 27 (2009), 75–84. MR2488989 Zbl 1165.5305110.1016/j.difgeo.2008.06.007Search in Google Scholar
[7] S. Deng, Homogeneous Finsler spaces. Springer 2012. MR2962626 Zbl 1253.5300210.1007/978-1-4614-4244-8Search in Google Scholar
[8] S. Deng, Z. Hou, The group of isometries of a Finsler space. Pacific J. Math. 207 (2002), 149–155. MR1974469 Zbl 1055.5305510.2140/pjm.2002.207.149Search in Google Scholar
[9] S. Deng, Z. Hou, Invariant Randers metrics on homogeneous Riemannian manifolds. J. Phys. A37 (2004), 4353–4360. MR2063598 Zbl 1049.8300510.1088/0305-4470/37/15/004Search in Google Scholar
[10] S. Deng, Z. Hou, Naturally reductive homogeneous Finsler spaces. Manuscripta Math. 131 (2010), 215–229. MR2574999 Zbl 1197.5309210.1007/s00229-009-0314-zSearch in Google Scholar
[11] Z. Dušek, O. Kowalski, S. v. Nikčević, New examples of Riemannian g.o. manifolds in dimension 7. Differential Geom. Appl. 21 (2004), 65–78. MR2067459 Zbl 1050.2201110.1016/j.difgeo.2004.03.006Search in Google Scholar
[12] C. S. Gordon, Homogeneous Riemannian manifolds whose geodesics are orbits. In: Topics in geometry, volume 20 of Progr. Nonlinear Differential Equations Appl., 155–174, Birkhäuser Boston, Boston, MA 1996. MR1390313 Zbl 0861.5305210.1007/978-1-4612-2432-7_4Search in Google Scholar
[13] Z. Hu, S. Deng, Homogeneous Randers spaces with isotropic S-curvature and positive flag curvature. Math. Z. 270 (2012), 989–1009. MR2892934 Zbl 1239.5309510.1007/s00209-010-0836-9Search in Google Scholar
[14] O. Kowalski, L. Vanhecke, Riemannian manifolds with homogeneous geodesics. Boll. Un. Mat. Ital. B (7)5 (1991), 189–246. MR1110676 Zbl 0731.5304610.1142/9789814439381_0003Search in Google Scholar
[15] D. Latifi, Homogeneous geodesics in homogeneous Finsler spaces. J. Geom. Phys. 57 (2007), 1421–1433. MR2289656 Zbl 1113.5304810.1016/j.geomphys.2006.11.004Search in Google Scholar
[16] D. Montgomery, H. Samelson, Transformation groups of spheres. Ann. of Math. (2)44 (1943), 454–470. MR8817 Zbl 0063.0407710.2307/1968975Search in Google Scholar
[17] Y. G. Nikonorov, Geodesic orbit Riemannian metrics on spheres. Vladikavkaz. Mat. Zh. 15 (2013), 67–76. MR3881073 Zbl 1293.53062Search in Google Scholar
[18] Y. G. Nikonorov, E. D. Rodionov, V. V. Slavskiĭ, Geometry of homogeneous Riemannian manifolds. Sovrem. Mat. Prilozh. no. 37, Geometriya (2006), 101–178. English translation: J. Math. Sci. (N.Y.)146 (2007), no. 6, 6313–6390. MR2568572 Zbl 1154.5302910.1007/s10958-007-0472-zSearch in Google Scholar
[19] G. Randers, On an asymmetrical metric in the fourspace of general relativity. Phys. Rev. (2)59 (1941), 195–199. MR3371 Zbl 0027.1810110.1103/PhysRev.59.195Search in Google Scholar
[20] C. Riehm, Explicit spin representations and Lie algebras of Heisenberg type. J. London Math. Soc. (2)29 (1984), 49–62. MR734990 Zbl 0542.1501010.1112/jlms/s2-29.1.49Search in Google Scholar
[21] M. Xu, Geodesic orbit spheres and constant curvature in Finsler geometry. Differential Geom. Appl. 61 (2018), 197–206. MR3856756 Zbl 1404.5309410.1016/j.difgeo.2018.07.002Search in Google Scholar
[22] Z. Yan, S. Deng, Finsler spaces whose geodesics are orbits. Differential Geom. Appl. 36 (2014), 1–23. MR3262894 Zbl 1308.5311410.1016/j.difgeo.2014.06.006Search in Google Scholar
[23] W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259 (1982), 351–358. MR661203 Zbl 0469.5304310.1007/BF01456947Search in Google Scholar
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