Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Examples of Finsler metrics without conjugate points: Metrics of revolution

Author: N. Zinov'ev
Translated by: the author
Original publication: Algebra i Analiz, tom 20 (2008), nomer 3.
Journal: St. Petersburg Math. J. 20 (2009), 361-379
MSC (2000): Primary 53B40
Published electronically: April 6, 2009
MathSciNet review: 2454452
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A large class of Finsler metrics without conjugate points on the torus that are far away from flat metrics is constructed.

References [Enhancements On Off] (What's this?)

  • [Arn] V. I. Arnol'd, Mathematical methods of classical mechanics, ``Nauka'', Moscow, 1989; English transl. of the 1974 Russian original, Grad. Texts in Math., vol. 60, Springer-Verlag, New York, 1992. MR 1037020 (93c:70001); MR 1345386 (96c:70001)
  • [Ban] V. Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems 10 (1989), 263-286. MR 1062758 (91j:58126)
  • [BBI] D. Yu. Burago, Yu. D. Burago, and S. V. Ivanov, A course in metric geometry, Moscow-Izhevsk, 2004; English transl., Grad. Stud. in Math., vol. 33, Amer. Math. Soc., Providence, RI, 2001. MR 1835418 (2002e:53053)
  • [Bu1] H. Busemann, Metrics on the torus without conjugate points, Bol. Soc. Mat. Mexicana 10 (1953), nos. 3-4, 1-18. MR 0065225 (16:399f)
  • [Bu2] -, The geometry of geodesics, Acad. Press, Inc., New York, NY, 1955. MR 0075623 (17:779a)
  • [Bur] D. Burago, Periodic metrics, Adv. Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 205-210. MR 1166203 (93c:53029)
  • [BuI] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), no. 3, 259-269. MR 1274115 (95h:53049)
  • [BF] A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian systems. Geometry, topology, classification, Udmurt. Univ., Izhevsk, 1999; English transl., Chapman and Hall/CRC, Boca Raton, FL, 2004. MR 1792190 (2002g:37071); MR 2036760 (2004j:37106)
  • [CK] C. B. Croke and B. Kleiner, On tori without conjugate points, Invent. Math. 120 (1995), 241-257. MR 1329041 (96j:53037)
  • [DNF] B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Modern geometry--methods and applications. Parts I-III, 5th ed., ``Editorial URSS'', Moscow, 2001; English transl. of 1st ed., Part I, Grad. Texts in Math., vol. 93, Springer-Verlag, New York, 1984; Part II, Grad. Texts in Math., vol. 104, Springer-Verlag, New York, 1985; Part III, Grad. Texts in Math., vol. 124, Springer-Verlag, New York, 1990. MR 0736837 (85a:53003); MR 0807945 (86m:53001); MR 1076994 (91j:55001)
  • [Fom] A. T. Fomenko, Symplectic geometry, Moskov. Gos. Univ., Moscow, 1988; English transl., Adv. Stud. in Contemp. Math., vol. 5, Gordon and Breach Sci. Publ., New York, 1988. MR 0964470 (90k:58082); MR 0994805 (90k:58065)
  • [H] Jens Heber, On the geodesic flow of tori without conjugate points, Math. Z. 216 (1994), 209-216. MR 1278421 (95d:58104)
  • [Sh] Z. Shen, Lectures on Finsler geometry, World Sci. Publ. Co., Singapore, 2001. MR 1845637 (2002f:53032)
  • [TF] V. V. Trofimov and A. T. Fomenko, Algebra and geometry of integrable Hamiltonian differential equations, Mathematics and its Applications, ``Faktorial'', Moscow; copublished with Udmurt. Univ., Izhevsk, 1995. (Russian) MR 1469742 (98f:58118)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 53B40

Retrieve articles in all journals with MSC (2000): 53B40

Additional Information

N. Zinov'ev
Affiliation: Sun Microsystems SPB, LLC, 10th Krasnoarmeyskaya Street, 22-A, St. Petersburg, 190103, Russia

Keywords: Hopf hypothesis, flat Finsler metrics, metric of revolution
Received by editor(s): February 8, 2008
Published electronically: April 6, 2009
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society