Examples of Finsler metrics without conjugate points: metrics of revolution

Zinov′ev, N.

doi:10.1090/S1061-0022-09-01052-8

Examples of Finsler metrics without conjugate points: metrics of revolution
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by N. Zinov′ev
Translated by: the author

St. Petersburg Math. J. 20 (2009), 361-379

DOI: https://doi.org/10.1090/S1061-0022-09-01052-8

Published electronically: April 6, 2009

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Abstract:

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

References

V. I. Arnol′d, Matematicheskie metody klassicheskoĭ mekhaniki, 3rd ed., “Nauka”, Moscow, 1989 (Russian). MR 1037020
Victor Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems 10 (1990), no. 2, 263–286. MR 1062758, DOI 10.1017/S014338570000554X
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
Herbert Busemann, Metrics on the torus without conjugate points, Bol. Soc. Mat. Mexicana 10 (1953), no. nos. 3-4, 1–18. MR 0065225
Herbert Busemann, The geometry of geodesics, Academic Press, Inc., New York, N.Y., 1955. MR 0075623
D. Yu. Burago, Periodic metrics, Representation theory and dynamical systems, Adv. Soviet Math., vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 205–210. MR 1166203
D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), no. 3, 259–269. MR 1274115, DOI 10.1007/BF01896241
A. V. Bolsinov and A. T. Fomenko, Integriruemye gamil′tonovy sistemy. 1, 2, Izdatel′skiĭ Dom “Udmurt⋅skiĭ Universitet”, Izhevsk, 1999 (Russian, with Russian summary). Geometriya, topologiya, klassifikatsiya. [Geometry, topology, classification]; Appendices 2 and 3 by V. V. Kalashnikov, Jr. MR 1792190
Christopher B. Croke and Bruce Kleiner, On tori without conjugate points, Invent. Math. 120 (1995), no. 2, 241–257. MR 1329041, DOI 10.1007/BF01241128
B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern geometry—methods and applications. Part I, Graduate Texts in Mathematics, vol. 93, Springer-Verlag, New York, 1984. The geometry of surfaces, transformation groups, and fields; Translated from the Russian by Robert G. Burns. MR 736837, DOI 10.1007/978-1-4684-9946-9
A. T. Fomenko, Simplekticheskaya geometriya, Moskov. Gos. Univ., Moscow, 1988 (Russian). Metody i prilozheniya. [Methods and applications]. MR 964470
Jens Heber, On the geodesic flow of tori without conjugate points, Math. Z. 216 (1994), no. 2, 209–216. MR 1278421, DOI 10.1007/BF02572318
Zhongmin Shen, Lectures on Finsler geometry, World Scientific Publishing Co., Singapore, 2001. MR 1845637, DOI 10.1142/9789812811622
V. V. Trofimov and A. T. Fomenko, Algebra i geometriya integriruemykh gamil′tonovykh differentsial′nykh uravneniĭ, Matematika i ee Prilozheniya. [Mathematics and its Applications], Izdatel′stvo “Faktorial”, Moscow; copublished with Izdatel′stvo Udmurt⋅skogo Universiteta, Izhevsk, 1995 (Russian, with English and Russian summaries). MR 1469742