Introduction

Summary

In this section we gather together papers on symplectic and contact geometry. Recall that a symplectic manifold (M,ω) is a smooth manifold M of even dimension 2n with a closed, nondegenerate, 2-form ω i.e dω = 0 and ωⁿ is nowhere zero. A contact structure is an odd-dimensional analogue; a contact manifold (V,H) is a pair consisting of a manifold V of odd dimension 2n + 1 with a field H of 2n-dimensional subspaces of the tangent bundle TV which is maximally non-integrable, in the sense that if α is a 1-form defining H, then dαⁿ ∧ α is non-zero (i.e. dα is non-degenerate on H).

In their different ways, all the articles in this section are motivated by the work of M. Gromov, and in particular by his paper [G2] on pseudo-holomorphic curves. Here the idea is to replace a complex manifold by an almost-complex manifold with a compatible symplectic structure, and to study the generalisations of the complex curves–defined by this almost-complex structure. The paper of McDuff below gives a direct application of this method by showing that a minimal 4-dimensional symplectic manifold containing an embedded, symplectic, copy of S² = CP¹ is either CP² or an S² bundle over a Riemann surface, with the symplectic form being non-degenerate on fibres. The uniqueness of the structure in the minimal case can be thought of as an example of rigidity.