no. 3
Finsler Conformal Lichnerowicz-Obata conjecture
[La conjecture de Lichnerowicz-Obata sur les transformations conformes des variétés Finslériennes]
Matveev, V. S.1 ; Rademacher, H.-B.2 ; Troyanov, M.3 ; Zeghib, A.4
1 Mathematisches Institut, Friedrich-Schiller Universität Jena 07737 Jena (Germany)
2 Mathematisches Institut, Universität Leipzig, 04081 Leipzig (Germany)
3 Section de Mathématiques, École Polytechnique Fédérale de Lausanne 1015 Lausanne (Switzerland)
4 UMPA, ENS-Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07 (France)
Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 937-949.

Nous démontrons une variante de la conjecture de Lichnerowicz-Obata sur les transformations conformes des variétés finslériennes. Plus précisément, un champ de vecteurs conforme complet et essentiel sur une variété finslérienne non-riemannienne, est un champ homothétique sur un espace vectoriel normé.

We prove the Finsler analog of the conformal Lichnerowicz-Obata conjecture showing that a complete and essential conformal vector field on a non-Riemannian Finsler manifold is a homothetic vector field of a Minkowski metric.

DOI : 10.5802/aif.2452
Classification : 58b20, 53c60
Keywords: Finsler metric, conformal transformation
Mot clés : métrique finslérienne, transformation conforme

Matveev, V. S. 1 ; Rademacher, H.-B. 2 ; Troyanov, M. 3 ; Zeghib, A. 4

1 Mathematisches Institut, Friedrich-Schiller Universität Jena 07737 Jena (Germany)
2 Mathematisches Institut, Universität Leipzig, 04081 Leipzig (Germany)
3 Section de Mathématiques, École Polytechnique Fédérale de Lausanne 1015 Lausanne (Switzerland)
4 UMPA, ENS-Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07 (France) 
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     title = {Finsler {Conformal} {Lichnerowicz-Obata} conjecture},
     journal = {Annales de l'Institut Fourier},
     pages = {937--949},
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Matveev, V. S.; Rademacher, H.-B.; Troyanov, M.; Zeghib, A. Finsler Conformal Lichnerowicz-Obata conjecture. Annales de l'Institut Fourier, Tome 59 (2009) no. 3, pp. 937-949. doi : 10.5802/aif.2452. https://aif.centre-mersenne.org/articles/10.5802/aif.2452/

[1] Akbar-Zadeh, H. Transformations infinitésimales conformes des variétés finslériennes compactes, Ann. Polon. Math., Volume 36 (1979), pp. 213-229 | MR | Zbl

[2] Alekseevskii, D. V. Groups of conformal transformations of Riemannian spaces, (russian) Mat. Sbornik, 89 (131), 1972 (engl.transl.) Math. USSR Sbornik, Volume 18 (1972), pp. 285-301 | MR | Zbl

[3] Alvarez Paiva, J. C. Some problems on Finsler geometry, Handbook of differential geometry, II, p. 1–33, Elsevier/North-Holland, Amsterdam, 2006 | MR | Zbl

[4] Bao, D.; Chern, S.- S.; Shen, Z. An Introduction to Riemann-Finsler Geometry, Grad. Texts Mathem., 200, Springer Verlag, New York, 2000 | MR | Zbl

[5] Benedetti, R.; Petronio, V. Lectures on Hyperbolic Geometry, Springer, Berlin, 1972 | Zbl

[6] Bryant, R. L. Projectively flat Finsler 2-spheres of constant curvature, Selecta Math., Volume 3 (1997), pp. 161-203 | DOI | MR | Zbl

[7] Burago, D.; Burago, Yu.; Ivanov, S. A course in metric geometry, Graduate Studies in Mathematics, 33, AMS, Providence, RI, 2001 | Zbl

[8] Ferrand, J. Le groupe des automorphismes conformes d’une variété de Finsler compacte, C. R. Acad.Sci, Paris, Sér. A-B 291, (1980) (A209-A210) | Zbl

[9] Ferrand, J. The action of conformal transformations on a Riemannian manifold, Math. Ann., Volume 304 (1996), pp. 277-291 | DOI | MR | Zbl

[10] Frances, C.; Tarquini, C. Autour du théorème de Ferrand-Obata, Ann. Gl. Anal. Geom., Volume 21 (2002), pp. 51-62 | DOI | MR | Zbl

[11] Kühnel, W.; Rademacher, H.-B. Liouville’s theorem in conformal geometry, J. Math. pures appl., Volume 88 (2007), pp. 251-260 | DOI | Zbl

[12] Kulkarni, R. S.; R. S. Kulkarni and U. Pinkall Conformal structures and Möbius structures, Conformal Geometry, p. 1–39, Aspects Math. E 12, Vieweg, Braunschweig, 1988 | MR | Zbl

[13] Lafontaine, J.; R. S. Kulkarni and U. Pinkall The theorem of Lelong-Ferrand andObata, Conformal Geometry, p. 93–103, Aspects Math. E 12, Vieweg, Braunschweig, 1988 | MR | Zbl

[14] Lie, S. Ueber Complexe, insbesondere Linien- und Kugel-Complexe mit Anwendung auf die Theorie partieller Differentialgleichungen, Math. Ann., Volume 5 (1872), pp. 145-246 | DOI | MR

[15] Liouville, J. Extension au cas des trois dimensions de la question du tracé géographique, Applications de l’analyse à la géométrie, Bachelier, Note VI, G. Monge, Paris, 1850, pp. 609-617

[16] Matveev, V. S. Lichnerowicz-Obata conjecture in dimension two, Comm. Math. Helv., Volume 81 (2005) no. 3, pp. 541-570 | DOI | MR | Zbl

[17] Matveev, V. S. Proof of projective Lichnerowicz-Obata conjecture, J. Diff. Geom., Volume 75 (2007), pp. 459-502 (arXiv:math/0407337) | MR | Zbl

[18] Obata, M. The conjectures about conformal transformations, J. Diff. Geom., Volume 6 (1971), pp. 247-258 | MR | Zbl

[19] Schoen, R. On the conformal and CR automorphism groups, Geom. Funct. Anal., Volume 5 (1995) no. 2, pp. 464-481 | DOI | MR | Zbl

[20] Shen, Z. Lectures on Finsler geometry, World Scientific, Singapore, 2001 | MR | Zbl

[21] Szabó, Z. I. Berwald metrics constructed by Chevalley’s polynomials (arXiv:math/0601522)

[22] Torrome, R. G. Average Riemannian structures associated with a Finsler structure (arXiv:math/0501058v5)

[23] Yoshimatsu, Y. On a theorem of Alekseevskii concerning conformal transformations, J.Math.Soc.Japan, Volume 28 (1976), pp. 278-289 | DOI | MR | Zbl

[24] Zeghib, A. On the conformal group of Finsler manifolds, preprint, 2005 (http://www.umpa.ens-lyon.fr/~zeghib/finsconf.pdf)

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