Lagrangian intersection under Legendrian deformations
type: Item
instance of: scholarly article
title: Lagrangian intersection under Legendrian deformations
zbMATH Open document ID: 0868.58035
DOI: 10.1215/S0012-7094-96-08509-9
author: Kaoru Ono
published in: Duke Mathematical Journal
publication date: 1997-08-21T00:00:00Z
review text: From Arnold's conjecture for Lagrangian submanifolds -- ``Let \(L\) be a Lagrangian submanifold in a symplectic manifold \((M,\omega)\). For a Hamiltonian isotopy \(\{\phi_t|0\leq t\leq 1\}\), the number of intersection points of \(L\) and \(\phi_1(L)\) is at least the cup-length of \(L\). Moreover, if they intersect transversally, this number is at least the sum of Betti numbers of \(L\). ``This paper deals with the following particular question: ``Let \(\pi:P\to M\) be a principal \(S^1\)-bundle of \((M,\omega)\), with a connection such that the curvature equals \(-\omega\) and \(\Lambda\) a Legendrian submanifold in \(P\). Does the number of intersection points of \(L\) and \(\pi\circ \psi_1(\Lambda)\) satisfy the same inequality as above? \((\{\psi_t\mid 0\leq t\leq 1\}\) is a contact isotopy group on \(P\)).'' The author shows that this number is at least the sum of \(\mathbb{Z}_2\)-Betti numbers of \(L\), provided that \(\pi_2(M,L)=0\), and \(L\) and \(\pi\circ \psi_1(\Lambda)\) intersect transversally.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 57R22
zbMATH DE Number: 949891
zbMATH Keywords: Lagrangian intersections
zbMATH Keywords: Legendrian deformations
reviewed by: Paulo R. Rodrigues
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 949891