A note on the space of Lagrangian submanifolds of a symplectic 4-manifold
type: Item
instance of: scholarly article
title: A note on the space of Lagrangian submanifolds of a symplectic 4-manifold
zbMATH Open document ID: 0986.53033
DOI: 10.1016/S0393-0440(00)00008-5
author: Tim Swift
published in: Journal of Geometry and Physics
publication date: 2002-06-12T00:00:00Z
review text: Let \(N\) be a connected manifold without boundary and let \(M\) be a compact oriented submanifold of \(N\). Then \(\text{Sub}(M,N)\) is defined as the manifold of all submanifolds of \(N\) which are of diffeomorphism type \(M\). Assuming then that \(M\) is a codimension two submanifold and that \(N\) admits a volume element \(\eta\), the author shows that \(\text{Sub}(M, N)\) admits a unique weakly symplectic form. This is then applied to the study of Lagrangian submanifolds in a symplectic 4-manifold. There, \(\text{Sub}_0(M,N)\) denotes the manifold of all Lagrangian submanifolds of \(N\) which are of the diffeomorphism type of a Lagrangian surface \(M\). It is then shown that the natural embedding \(\text{Sub}_0(M, N)\to \text{Sub}(M, N)\) is coisotropic embedding. Moreover, it is also shown that this map is Lagrangian if and only if \(M\) is diffeomorphic with \(S^2\).
reviewed by: Luc Vrancken
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 58D10
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 1523515
zbMATH Keywords: symplectic 4-manifold
zbMATH Keywords: Lagrangian submanifold
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1523515