Rigidity of magnetic flows for compact surfaces

José Barbosa Gomes; Rafael O. Ruggiero

doi:10.1016/j.crma.2008.01.011

Topology/Dynamical Systems

Rigidity of magnetic flows for compact surfaces
[Rigidité des flots magnétiques sur des surfaces compactes]

Présenté par : Étienne Ghys

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

Comptes Rendus. Mathématique, Volume 346 (2008) no. 5-6, pp. 313-316.

Résumé
Abstract

Soit $ϕ_{t} : T_{1} M \to T_{1} M$ le flot magnétique du pair $(g, Ω)$ . Nous demonstrons que si $ϕ_{t}$ preserve un feuilletage $C^{2, 1}$ 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 $ϕ_{t} : T_{1} M \to T_{1} M$ be the magnetic flow of the pair $(g, Ω)$ . We show that if $ϕ_{t}$ preserves a $C^{2, 1}$ 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 : 2007-12-26
Accepté le : 2008-01-07
Publié le : 2008-02-15

DOI : 10.1016/j.crma.2008.01.011

Affiliations des auteurs :

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

@article{CRMATH_2008__346_5-6_313_0,
     author = {Jos\'e Barbosa Gomes and Rafael O. Ruggiero},
     title = {Rigidity of magnetic flows for compact surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {313--316},
     publisher = {Elsevier},
     volume = {346},
     number = {5-6},
     year = {2008},
     doi = {10.1016/j.crma.2008.01.011},
     language = {en},
}

TY  - JOUR
AU  - José Barbosa Gomes
AU  - Rafael O. Ruggiero
TI  - Rigidity of magnetic flows for compact surfaces
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 313
EP  - 316
VL  - 346
IS  - 5-6
PB  - Elsevier
DO  - 10.1016/j.crma.2008.01.011
LA  - en
ID  - CRMATH_2008__346_5-6_313_0
ER  -

%0 Journal Article
%A José Barbosa Gomes
%A Rafael O. Ruggiero
%T Rigidity of magnetic flows for compact surfaces
%J Comptes Rendus. Mathématique
%D 2008
%P 313-316
%V 346
%N 5-6
%I Elsevier
%R 10.1016/j.crma.2008.01.011
%G en
%F CRMATH_2008__346_5-6_313_0

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/

Bibliographie
Cité par

[1] M.L. Bialy Rigidity for periodic magnetic fields, Ergodic Theory Dynam. Systems, Volume 20 (2000), pp. 1619-1626

[2] E. Ghys Flots d'Anosov dont les feuilletages stables sont différentiables, Ann. Sci. École Norm. Sup., Volume 20 (1987), pp. 251-270

[3] E. Ghys Rigidité différentiable des groupes Fuchsiens, Publ. Math. I.H.E.S., Volume 78 (1993), pp. 163-185

[4] J.B. Gomes; R.O. Ruggiero Rigidity of surfaces whose geodesic flows preserve smooth foliations of codimension 1, Proc. Amer. Math. Soc., Volume 135 (2007), pp. 507-515

[5] S. Grognet Flots magnétiques en courbure négative, Ergodic Theory Dynam. Systems, Volume 19 (1999), pp. 413-436

[6] S. Hurder; A. Katok Differentiability, rigidity and Godbillon–Vey classes for Anosov flows, Publ. Math. I.H.E.S., Volume 72 (1990), pp. 5-61

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

[8] G.P. Paternain Regularity of weak foliations for thermostats, Nonlinearity, Volume 20 (2007), pp. 87-104

[9] G.P. Paternain; M. Paternain On Anosov energy levels of convex Hamiltonian systems, Math. Z., Volume 217 (1994), pp. 367-376

[10] G.P. Paternain; M. Paternain Anosov geodesic flows and twisted symplectic structures (F. Ledrappier; J. Lewowics; S. Newhouse, eds.), Pitman Research Notes in Mathematics Series, vol. 362, Addison-Wesley, Longman, Harlow, 1996, pp. 132-145

[11] G. Raby Invariance des classes de Godbillon–Vey par $C^{1}$ -diffeomorphismes, Ann. Inst. Fourier, Volume 38 (1998) no. 1, pp. 205-213

Cité par Sources :

Commentaires - Politique