Comptes Rendus
no. 1
Differential geometry
There exist no locally symmetric Finsler spaces of positive or negative flag curvature
[Il n'existe pas d'espace de Finsler localement symétrique de courbure de drapeau positive ou négative]

Présenté par : Étienne Ghys

Vladimir S. Matveev1
1 Mathematisches Institut, Friedrich-Schiller Universität Jena, 07737 Jena, Germany
Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 81-83.

Nous montrons que les résultats de Foulon [5,6] et de Kim [7] (et indépendamment, de Deng et Hou [4]) sur l'inexistence de métriques de Finsler localement symétriques, de courbure de drapeau positive ou négative, sont en fait locaux.

We show that the results of Foulon [5,6] and Kim [7] (independently, of Deng and Hou [4]) about the nonexistence of locally symmetric Finsler metrics of positive or negative flag curvature are in fact local.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.10.022
Vladimir S. Matveev 1

1 Mathematisches Institut, Friedrich-Schiller Universität Jena, 07737 Jena, Germany 
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Vladimir S. Matveev. There exist no locally symmetric Finsler spaces of positive or negative flag curvature. Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 81-83. doi : 10.1016/j.crma.2014.10.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.10.022/

[1] D. Bao; C. Robles Ricci and Flag Curvatures in Finsler Geometry, Riemann–Finsler Geometry MSRI Publications, vol. 50, 2004

[2] D. Bao; S.S. Chern; Z. Shen An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag Inc., New York, 2000

[3] C. Chevalley Invariants of finite groups generated by reflections, Amer. J. Math., Volume 77 (1955), pp. 778-782

[4] S. Deng; Z. Hou On symmetric Finsler spaces, Isr. J. Math., Volume 162 (2007), pp. 197-219

[5] P. Foulon Locally symmetric Finsler spaces in negative curvature, C. R. Acad. Sci. Paris, Ser. I, Volume 324 (1997) no. 10, pp. 1127-1132

[6] P. Foulon Curvature and global rigidity in Finsler manifolds, Houst. J. Math., Volume 28 (2002) no. 2, pp. 263-292

[7] C.-W. Kim Locally symmetric positively curved Finsler spaces, Arch. Math., Volume 880 (2007), pp. 378-384

[8] V.S. Matveev; M. Troyanov The Binet–Legendre metric in Finsler geometry, Geom. Topol., Volume 16 (2012), pp. 2135-2170

[9] R.S. Palais; C.-L. Terng Critical Point Theory and Submanifold Geometry, Lecture Notes in Mathematics, vol. 1353, Springer-Verlag, New York, 1988

[10] J. Simons On transitivity of holonomy systems, Ann. of Math. (2), Volume 76 (1962), pp. 213-234

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