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Geodesics in Randers spaces of constant curvature

Author: Colleen Robles
Journal: Trans. Amer. Math. Soc. 359 (2007), 1633-1651
MSC (2000): Primary 53B40, 53C60
Published electronically: October 16, 2006
MathSciNet review: 2272144
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Abstract: Geodesics in Randers spaces of constant curvature are classified.

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  • [AZ88] H. Akbar-Zadeh, Sur les espaces de Finsler á courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), 281-322.MR 1052466 (91f:53069)
  • [A55] L. Auslander, On curvature in Finsler geometry, Trans. of Amer. Math. Soc. 79 (1955), 378-388. MR 0071833 (17:190d)
  • [B78] W. Ballmann, Der Satz von Lusternik und Schnirelmann (German), Beiträge zur Differentialgeometrie, Heft 1, 1-25. Bonner Math. Schriften, 102, Univ. Bonn, Bonn, 1978. MR 0520178 (80d:58017)
  • [BL04] V. Bangert and Y. Long, Multiple closed geodesics on Finsler 2-spheres and a conjecture of D. V. Anosov, in preparation.
  • [BCS00] D. Bao, S.S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, 200, Springer, 2000. MR 1747675 (2001g:53130)
  • [BR04] D. Bao and C. Robles, On Ricci and flag curvatures in Finsler geometry, in Some perspectives in Finsler geometry, MSRI Publications 50, Cambridge University Press, 2004. MR 2132660 (2005k:53124)
  • [BRS04] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds, J. Diff. Geom. 66 (2004), 391-449. MR 2106471 (2005k:58023)
  • [BS02] D. Bao and Z. Shen, Finsler metrics of constant positive curvature on the Lie group $ S^3$, J. London Math. Soc. 66 (2002), 453-467. MR 1920414 (2003g:53129)
  • [BF02] A. Bejancu and H. Farran, Finsler metrics of positive constant flag curvature on Sasakian space forms, Hokkaido Math. J. 31(2) (2002), 459-468. MR 1914971 (2003f:53133)
  • [BF03] A. Bejancu and H. Farran, Randers manifolds of positive constant curvature, Int'l. J. Math. & Math'l. Sc. 18 (2003), 1155-1165. MR 1978413 (2004d:53090)
  • [Br96] R. Bryant, Finsler structures on the 2-sphere satisfying $ K=1$, Cont. Math. 196 (1996), 27-41. MR 1403574 (97e:53128)
  • [Br97] R. Bryant, Projectively flat Finsler 2-spheres of constant curvature, Selecta Math. (N.S.) 3(2) (1997), 161-203. MR 1466165 (98i:53101)
  • [Br04] R. Bryant, Geodesically reversible Finsler 2-spheres of constant curvature, arXiv:math.DG/0407514.
  • [C99] C. Carathéodory: Calculus of Variations and Partial Differential Equations of the First Order, AMS Chelsea Publishing, 1999. MR 1279593 (96e:49001)
  • [F29] P. Funk, Über Geometrien, bei denen die Geraden die Kürzesten sind, Math. Ann. 101 (1929), 226-237. MR 1512527
  • [G89] M. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71-111. MR 0979601 (90a:53050)
  • [K73] A. B. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems, Math. USSR Izvestija (3) 7 (1973), 535-571.MR 0331425 (48:9758)
  • [LS29] L. Lusternik and L. Schnirelmann, Sur le problème de trois géodésiques fermées sur les surfaces de genre 0, C. R. Acad. Sci. Paris 189 (1929), 269-271.
  • [LS30] L. Lusternik and L. Schnirelmann, Topological methods in the calculus of variations, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (1930) [Russian]; Méthodes topologiques dans les problèmes variationelles, 1 $ ^{\hbox{re}}$ partie, Actualité Sci. Industr. 188 (1934).
  • [O83] T. Okada, On models of projectively flat Finsler spaces of constant negative curvature, Tensor, N.S. 40 (1983), 117-124. MR 0837784 (87c:53124)
  • [R41] G. Randers, On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59 (1941), 195-199. MR 0003371 (2:208a)
  • [Ra61] A. Rapcsák, Über die bahntreuen Abbidungen metrischer Räume, Publ. Math., Debrecen 8 (1961), 285-290. MR 0138079 (25:1526)
  • [S96] Z. Shen, Finsler spaces of constant positive curvature, in Finsler Geometry, Contemporary Math. 196 (1996), 83-92. MR 1403580 (97m:53120)
  • [S02] Z. Shen, Finsler metrics with $ K=0$ and $ S=0$, Canadian J. Math. 55 (2003), 112-132. MR 1952328 (2004e:53112)
  • [Z31] E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher Windverteilung, Z. Angew. Math. Mech. 11 (1931), 114-124.
  • [Zi82] W. Ziller, Geometry of the Katok examples, Ergod. Th. & Dynam. Sys. 3 (1982), 135-157. MR 0743032 (86g:58036)

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Additional Information

Colleen Robles
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Keywords: Finsler geometry, Randers metric, constant curvature, geodesics, Zermelo navigation, infinitesimal homothety
Received by editor(s): January 19, 2005
Published electronically: October 16, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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