On Lagrangian embeddings of parallelizable manifolds
type: Item
instance of: scholarly article
zbMATH Open document ID: 1277.57023
DOI: 10.1142/S0129167X13500730
author: Naohiko Kasuya
author: Toru Yoshiyasu
published in: International Journal of Mathematics
publication date: 2013-10-25T00:00:00Z
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D15
zbMATH DE Number: 6220075
zbMATH Keywords: symplectic structures
zbMATH Keywords: Lagrangian submanifolds
zbMATH Keywords: \(h\)-principle
zbMATH Keywords: Kervaire semi-characteristic
MaRDI profile type: Publication
cites work: On the non-existence of elements of Hopf invariant one
cites work: Normal bundles of double dimension immersions, double points of Lagrangian immersions, and totally real embeddings
cites work: Pseudo holomorphic curves in symplectic manifolds
cites work: Some remarks on Lagrangian imbeddings
cites work: Courbure intégrale généralisée et homotopie
cites work: Relative Characteristic Classes
cites work: Q5461285
cites work: Q753156
cites work: A classification of smooth embeddings of 3-manifolds in 6-space
title: On Lagrangian embeddings of parallelizable manifolds
review text: In this paper, the authors study the problem of realizing an \(n\)-manifold \(M^n\) as a Lagrangian submanifold in the \(2n\)-dimensional Euclidean space \(\mathbb R^{2n}\) with a certain symplectic structure. To obtain a Lagrangian embedding of an \(n\)-manifold in \(\mathbb R^{2n}\) it is sufficient to embed its cotangent bundle in \(\mathbb R^{2n}\) and extend its symplectic structure to \(\mathbb R^{2n}\). By using Gromov's \(h\)-principle for symplectic structures on an open manifold and the Kervaire semi-characteristic, the authors prove that for any closed parallelizable \(n\)-manifold \(M^n\), if the dimension \(n\neq 7\), or if \(n=7\) and the Kervaire semi-characteristic \(\chi_{\frac12}(M^7)\) is zero, then \(M^n\) can be embedded in the Euclidean space \(\mathbb R^{2n}\) with a certain symplectic structure as a Lagrangian submanifold. It turns out that for \(n=2\), the only closed parallelizable \(2\)-manifold is the \(2\)-torus and for \(n=3\), any closed orientable \(3\)-manifold is parallelizable. There are infinitely many isotopy classes of embeddings of the \(2\)-torus in the \(4\)-dimensional Euclidean space. For \(n\geq 3\), there is a surjection from the set of isotopy classes of embeddings of \(M^n\) in the \(2n\)-dimensional Euclidean space \(\mathbb R^{2n}\) to the homology group \(H_1(M^n;\mathbb Z)\) if \(n\) is odd, and to \(H_1(M^n;\mathbb Z_2)\) if \(n\) is even. By the results of Gromov and Fukaya, the authors' result gives rise to symplectic structures of \(\mathbb R^{2n}\) \((n\geq 3)\) which are not conformally equivalent to open domains in standard ones.
reviewed by: Andrew Bucki
description: scientific article; zbMATH DE number 6220075