The Reeb foliation arises as a family of Legendrian submanifolds at the end of a deformation of the standard $ {S}^{3}$ in $ {S}^{5}$

Atsuhide Mori

doi:10.1016/j.crma.2012.01.001

Differential Geometry

The Reeb foliation arises as a family of Legendrian submanifolds at the end of a deformation of the standard

S^{3}

in

S^{5}

[Le feuilletage de Reeb se réalise comme une famille de sous-variétés legendriennes à lʼaboutissement dʼune déformation dʼune sphère

S^{3}

canonique dans

S^{5}

]

Présenté par : Jean-Pierre Demailly

Atsuhide Mori ¹

¹ Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 67-70.

Abstract (VO)
Résumé

We realize the Reeb foliation of $S^{3}$ as a family of Legendrian submanifolds of the unit $S^{5} \subset C^{3}$ . Moreover, we construct a deformation of the standard contact $S^{3}$ in $S^{5}$ , via a family of contact submanifolds, into this realization.

Nous réalisons le feuilletage de Reeb comme une famille de sous-variétés legendriennes de la sphère unité $S^{5}$ dans $C^{3}$ . Par ailleurs, nous construisons une déformation de la structure de contact canonique $S^{3}$ dans $S^{5}$ via une famille de sous-variétés de contact, aboutissant au feuilletage ainsi réalisé.

Reçu le : 2010-09-10
Accepté le : 2012-01-03
Publié le : 2012-01-01

DOI : 10.1016/j.crma.2012.01.001

Affiliations des auteurs :

Atsuhide Mori ¹

¹ Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan

@article{CRMATH_2012__350_1-2_67_0,
     author = {Atsuhide Mori},
     title = {The {Reeb} foliation arises as a family of {Legendrian} submanifolds at the end of a deformation of the standard $ {S}^{3}$ in $ {S}^{5}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {67--70},
     publisher = {Elsevier},
     volume = {350},
     number = {1-2},
     year = {2012},
     doi = {10.1016/j.crma.2012.01.001},
     language = {en},
}

TY  - JOUR
AU  - Atsuhide Mori
TI  - The Reeb foliation arises as a family of Legendrian submanifolds at the end of a deformation of the standard $ {S}^{3}$ in $ {S}^{5}$
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 67
EP  - 70
VL  - 350
IS  - 1-2
PB  - Elsevier
DO  - 10.1016/j.crma.2012.01.001
LA  - en
ID  - CRMATH_2012__350_1-2_67_0
ER  -

%0 Journal Article
%A Atsuhide Mori
%T The Reeb foliation arises as a family of Legendrian submanifolds at the end of a deformation of the standard $ {S}^{3}$ in $ {S}^{5}$
%J Comptes Rendus. Mathématique
%D 2012
%P 67-70
%V 350
%N 1-2
%I Elsevier
%R 10.1016/j.crma.2012.01.001
%G en
%F CRMATH_2012__350_1-2_67_0

Atsuhide Mori. The Reeb foliation arises as a family of Legendrian submanifolds at the end of a deformation of the standard $ {S}^{3}$ in $ {S}^{5}$. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 67-70. doi : 10.1016/j.crma.2012.01.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.01.001/

Bibliographie
Cité par

[1] J. Alexander A lemma on systems of knotted curves, Proc. Nat. Acad. Sci. USA, Volume 9 (1923), pp. 93-95

[2] J. Etnyre Contact structures on 3-manifolds are deformations of foliations, Int. Math. Res. Notices, Volume 14 (2007), pp. 775-779

[3] E. Giroux, Géometrie de contact de la dimension trois vers les dimensions supérieures, in: Proc. ICM2002, Beijing, vol. II, pp. 405–414.

[4] D. Martínez Torres Contact embeddings in standard contact spheres via approximately holomorphic geometry, J. Math. Sci. Univ. Tokyo, Volume 18 (2011), pp. 139-154

[5] A. Mori A note on Thurston–Winkelnkemperʼs construction of contact forms on 3-manifolds, Osaka J. Math., Volume 39 (2002), pp. 1-11

[6] A. Mori Global models of contact forms, J. Math. Sci. Univ. Tokyo, Volume 11 (2004), pp. 447-454

[7] A. Mori Reeb foliations on $S^{5}$ and contact 5-manifolds violating the Thurston–Bennequin inequality, 2009 (preprint) | arXiv

[8] W. Thurston; H. Winkelnkemper On the existence of contact forms, Proc. AMS, Volume 52 (1975), pp. 345-347

Naohiko Kasuya Contact Structures, Non-Kähler Complex Surfaces and Strongly Pseudoconcave Surfaces (2025), p. 83 | DOI:10.1007/978-981-96-3002-8_6
Atsuhide Mori A Note on Mitsumatsu’s Construction of a Leafwise Symplectic Foliation, International Mathematics Research Notices, Volume 2019 (2019) no. 22, p. 6933 | DOI:10.1093/imrn/rnx321
John B. Etnyre; Ryo Furukawa Braided embeddings of contact 3-manifolds in the standard contact 5-sphere, Journal of Topology, Volume 10 (2017) no. 2, pp. 412-446 | DOI:10.1112/topo.12014 | Zbl:1377.53106
Naohiko Kasuya On the links of simple singularities, simple elliptic singularities and cusp singularities, Demonstratio Mathematica, Volume 48 (2015) no. 2, pp. 289-312 | DOI:10.1515/dema-2015-0021 | Zbl:1322.32021
Naohiko Kasuya The canonical contact structure on the link of a cusp singularity, Tokyo Journal of Mathematics, Volume 37 (2014) no. 1, pp. 1-20 | DOI:10.3836/tjm/1406552427 | Zbl:1307.57016

Cité par 5 documents. Sources : Crossref, zbMATH

Commentaires - Politique