Comptes Rendus
no. 5-6
Topology/Dynamical Systems
Rigidity of magnetic flows for compact surfaces
[Rigidité des flots magnétiques sur des surfaces compactes]

Présenté par : Étienne Ghys

José Barbosa Gomes1  ; Rafael O. Ruggiero2
1 Universidade Federal de Juiz de Fora, Dep. Matemática, Campus Universitário, Juiz de Fora, MG, Brasil 36036-330
2 Pontifícia Universidade Católica do Rio de Janeiro – PUC-Rio, Dep. Matemática, Rua Marquês de São Vicente, 225 – Gávea, Rio de Janeiro, RJ, Brasil 22453-900
Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 313-316.

Soit ϕt:T1M→T1M le flot magnétique du pair (g,Ω). Nous demonstrons que si ϕt preserve un feuilletage C2,1 de codimension 1, alors la courbure de (M,g) est une constante non positive et la forme Ω est le produit d'une constante par la forme d'aire de (M,g).

Let ϕt:T1M→T1M be the magnetic flow of the pair (g,Ω). We show that if ϕt preserves a C2,1 codimension one foliation then (M,g) has constant, nonpositive Gaussian curvature and Ω is a constant multiple of the area form of (M,g). So if the genus of M is greater than one, the flow is either Anosov or conjugate to a horocycle flow. If M is a torus, the flow is actually geodesic and flat.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.01.011
José Barbosa Gomes 1 ; Rafael O. Ruggiero 2

1 Universidade Federal de Juiz de Fora, Dep. Matemática, Campus Universitário, Juiz de Fora, MG, Brasil 36036-330
2 Pontifícia Universidade Católica do Rio de Janeiro – PUC-Rio, Dep. Matemática, Rua Marquês de São Vicente, 225 – Gávea, Rio de Janeiro, RJ, Brasil 22453-900 
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José Barbosa Gomes; Rafael O. Ruggiero. Rigidity of magnetic flows for compact surfaces. Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 313-316. doi : 10.1016/j.crma.2008.01.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.01.011/

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[7] G.P. Paternain Magnetic rigidity of horocycle flows, Pacific J. Math., Volume 225 (2006) no. 2, pp. 301-323

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