no. 13-14
Differential Geometry
The length of a shortest geodesic loop
[La longueur d'un lacet géodésique]

Présenté par : Marcel Berger

Rademacher, Hans-Bert1
1 Universität Leipzig, Mathematisches Institut, 04081 Leipzig, Germany
Comptes Rendus. Mathématique, Tome 346 (2008) no. 13-14, pp. 763-765.

On donne une borne inférieure pour la longueur d'un lacet géodésique non-triviale sur une variété compacte et simplement connexe munie d'une métrique de Finsler non-reversible de courbure positive. Harris et Paternain utilisent cette éstimée dans leur récent article afin de donner und charactérisation géométrique des métriques de Finsler à convexité dynamique sur la sphère de dimension 2.

We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper to give a geometric characterization of dynamically convex Finsler metrics on the 2-sphere.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.06.001
Rademacher, Hans-Bert 1

1 Universität Leipzig, Mathematisches Institut, 04081 Leipzig, Germany 
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Rademacher, Hans-Bert. The length of a shortest geodesic loop. Comptes Rendus. Mathématique, Tome 346 (2008) no. 13-14, pp. 763-765. doi : 10.1016/j.crma.2008.06.001. http://www.numdam.org/articles/10.1016/j.crma.2008.06.001/

[1] A. Harris, G. Paternain, Dynamically convex Finsler metrics and J-holomorphic embedding of asymptotic cylinders, Ann. Global Anal. Geom., in press, | DOI

[2] Hofer, H.; Wysocki, K.; Zehnder, E. The dynamics of three-dimensional strictly convex energy surfaces, Ann. Math. (2), Volume 148 (1998), pp. 197-289

[3] Klingenberg, W. Contributions to Riemannian geometry in the large, Ann. Math. (2), Volume 69 (1959), pp. 654-666

[4] Long, Y. Multiplicity and stability of closed geodesics on Finsler 2-spheres, J. Eur. Math. Soc., Volume 8 (2006), pp. 341-353

[5] Rademacher, H.B. A Sphere Theorem for non-reversible Finsler metrics, Math. Ann., Volume 328 (2004), pp. 373-387

[6] Rademacher, H.B. Non-reversible Finsler metrics of positive curvature (Bao, D.; Bryant, R.; Chern, S.S.; Shen, Z., eds.), A Sampler of Riemann–Finsler Geometry, MSRI Series, vol. 50, Cambridge Univ. Press, 2004, pp. 261-302

[7] Rademacher, H.B. Existence of closed geodesics on positively curved Finsler manifolds, Ergod. Theory Dynam. Systems, Volume 27 (2007), pp. 957-969

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