×

Some properties of Lagrangian embeddings. (Quelques propriétés des plongements Lagrangiens.) (French) Zbl 0751.58010

In the picture of the cotangent bundle the author presents some new properties of the Lagrangian submanifolds and of symplectic structures. I want to mention here two of them which seem to be very interesting: 1) Under some technical restrictions each \(C^ 0\) limit of Lagrangian submanifolds is a Lagrangian one. 2) The space of symplectic structures on the cotangent bundle is infinite dimensional.

MSC:

53D12 Lagrangian submanifolds; Maslov index
57R40 Embeddings in differential topology
57R52 Isotopy in differential topology
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] M. Audin , Fibrés normaux d’immersions en dimension double, points doubles d’immersions lagrangiennes et plongements totalement réels , Comment. Math. Helv. 63 ( 1988 ), 593-623. MR 89m:57032 | Zbl 0666.57024 · Zbl 0666.57024
[2] D. Bennequin , Caustique mystique , Séminaire Bourbaki n^\circ 635 (novembre 1984 ), Astérisque 133-134, 19-56. Numdam | Zbl 0632.58011 · Zbl 0632.58011
[3] I. Ekeland , H. Hofer , Symplectic topology and Hamiltonian dynamics , Math. Z. 200 ( 1989 ), 355-378. Partie II, Math. Z. 201 ( 1990 ). Article | MR 90a:58046 | Zbl 0641.53035 · Zbl 0641.53035
[4] Ya. M. Eliashberg , A theorem on the structure of wave fronts and its applications in symplectic topology , Funct. Anal. Appl. 21 ( 1987 ), 227-232. MR 88j:58033 | Zbl 0655.58015 · Zbl 0655.58015
[5] Ya. M. Eliashberg , préprint Stanford Univ. 1990 .
[6] A. Floer , H. Hofer , en préparation.
[7] M. Gromov , Pseudo-holomorphic curves in symplectic manifolds , Invent. Math. 82 ( 1985 ), 307-347. MR 87j:53053 | Zbl 0592.53025 · Zbl 0592.53025
[8] M. Gromov , Partial differential relations , Springer Ergebnisse 9 (nouvelle série), 1986 . MR 90a:58201 | Zbl 0651.53001 · Zbl 0651.53001
[9] M. Gromov , Symplectic topology, hard and soft , Int. Congress Math. Berkeley 1986 , Amer. Math. Soc., Providence, 1987 , p. 81-98. Zbl 0664.53016 · Zbl 0664.53016
[10] F. Lalonde et J.-C. Sikorav , Sous-variétés lagrangiennes des fibrés cotangents , à paraître dans Comment. Math. Helv. Zbl 0759.53022 · Zbl 0759.53022
[11] F. Laudenbach , Les formes différentielles de degré 1 non singulières : classes d’homotopie de leurs noyaux , Comment. Math. Helv. 51 ( 1976 ), 447-464. MR 55 #1360 | Zbl 0354.58001 · Zbl 0354.58001
[12] J.-P. Otal , Le spectre marqué des surfaces à courbure négative , Ann. of Math. ( 1990 ). MR 91c:58026 | Zbl 0699.58018 · Zbl 0699.58018
[13] L. Polterovich , The Maslov class of Lagrange surfaces and Gromov’s pseudo-holomorphic curves , à paraître dans Trans. Amer. Math. Soc. Zbl 0719.53016 · Zbl 0719.53016
[14] L. Polterovich , Monotone Lagrange submanifolds of linear spaces and the Maslov class in cotangent bundles , à paraître dans Math. Zeit. Article | Zbl 0703.58018 · Zbl 0703.58018
[15] L. Polterovich , The surgery of Lagrange submanifolds , préprint 1990 . · Zbl 0754.57027
[16] L. Polterovich , The Maslov class rigidity and non existence of Lagrangian embedding , préprint IHES 1990 .
[17] J.-C. Sikorav , Classes d’homotopie des formes fermées non singulières , C. R. Acad. Sci. Paris 294 ( 1982 ), 413-416. MR 83k:58004 | Zbl 0514.57012 · Zbl 0514.57012
[18] J.-C. Sikorav , Sur un problème de disjonction par isotopie symplectique , Ann. Ec. Norm. Sup. Paris 57 ( 1986 ), 543-552. Numdam | MR 88d:58045 | Zbl 0621.58018 · Zbl 0621.58018
[19] J.-C. Sikorav , Problèmes d’intersections et de points fixes en géométrie hamiltonienne , Comm. Math. Helv. 62 ( 1987 ), 61-72. MR 88g:58067 | Zbl 0684.58015 · Zbl 0684.58015
[20] J.-C. Sikorav , Homologie de Novikov , in : Thèse d’Etat, Université Paris-Sud Orsay, 1987 .
[21] J.-C. Sikorav , Rigidité symplectique dans le cotangent de Tn , Duke Math. J. 59 ( 1989 ), 227-231. MR 91e:58063 | Zbl 0697.53035 · Zbl 0697.53035
[22] C. Viterbo , A new obstruction to embedding Lagrangian tori , Invent. Math. 100 ( 1990 ), 301-320. MR 91d:58085 | Zbl 0727.58015 · Zbl 0727.58015
[23] C. Viterbo , Plongements lagrangiens et capacités symplectiques de tores dans \Bbb R2n , C.R.Acad.Sci. Paris 311 Série I ( 1990 ), 487-490. MR 92c:58041 | Zbl 0711.58012 · Zbl 0711.58012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.