A note on the space of Lagrangian submanifolds of a symplectic 4-manifold (Q1583937)

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 \(S^2\).