This chapter contains a brief introduction to nilmanifolds, and a discussion of Künneth and related structures on nilmanifolds. Nilmanifolds are homogeneous spaces for nilpotent Lie groups, and for them the discussions of geometric structures can often be reduced to the consideration of left-invariant structures. Left-invariant structures in turn arise from the corresponding linear structures on the Lie algebra, and these linear structures are usually much more tractable than arbitrary geometric structures on smooth manifolds. The nilmanifolds of abelian Lie groups are just tori, so that in some sense nilmanifolds are the simplest generalisations of tori.

We do not give a systematic treatment of nilmanifolds here, but focus on providing a few explicit examples of Künneth structures, of hypersymplectic structures, and of Anosov symplectomorphisms in this setting. For more information on topics from the theory of nilmanifolds that we treat rather breezily, we refer to the books by Gorbatsevich, Onishchik and Vinberg [GOV-97] and by Knapp [Kna-96].

In this chapter we discuss the linear algebra of symplectic vector spaces and symplectic vector bundles. To prepare the ground for the discussion of Künneth structures on manifolds in later chapters we introduce linear Künneth structures on vector bundles, and we work out consequences of the existence of Künneth structures in terms of characteristic classes.

The earlier parts of this chapter contain standard material that some readers may be able to skip. There is a substantial overlap, for example, with Chapter 2 of the book of McDuff-Salamon [McS-95]. The later parts contain some important results that are used throughout the book. While not original, these results clarify some of the folklore revolving around symplectic vector bundles and their Lagrangian subbundles. Our reference for the theory of characteristic classes is Milnor-Stasheff [MS-74].

In this chapter we introduce foliations and discuss some fundamental examples. We characterise the integrability of subbundles of tangent bundles in terms of both flatness and torsion-freeness of suitable affine connections. In the final section we discuss the simultaneous integrability of complementary distributions making up an almost product structure.

We introduce Bott connections in general, and we apply them to Lagrangian foliations in particular. This leads to a proof of Weinstein’s characterisation of affinely flat manifolds as leaves of Lagrangian foliations. We also prove a Darboux theorem for pairs consisting of a symplectic structure together with a Lagrangian foliation.

In this chapter we discuss Künneth geometry in real dimension four. Since in dimension two Künneth geometry is essentially Lorentz geometry, dimension four is really the first interesting case. For at least two reasons, it is also a very special case. First, it is possible to classify almost Künneth structures in terms of classical invariants. Second, four-dimensional symplectic geometry is very subtle, and symplectic structures in this dimension are constrained by their relation with Seiberg-Witten gauge theory. We will see that this makes it likely that Künneth four-manifolds may be classified, although we do not achieve that goal here, except in the hypersymplectic case.

Throughout this chapter we will use not only the material developed in earlier chapters of this book, but also the tools of modern four-dimensional geometry and topology. In particular we will use results from gauge theory. A good reference for both the basics of four-dimensional differential topology and results from Donaldson theory is the book by Donaldson and Kronheimer [DK-90]. In fact, very little Donaldson theory will be used in this chapter. We will make more use of results from Seiberg-Witten theory, for which we refer to the book by Morgan [Mor-96] and the second author’s Bourbaki lecture [Kot-97a] on Taubes’s work.

In this chapter we discuss the curvature of the Künneth connection. First we work out some general properties of the curvature tensor, then we prove a theorem showing that the curvature is the precise obstruction for the validity of the simplest possible Darboux theorem for Künneth structures. We then present some examples of vanishing and non-vanishing curvature, and we work out the Ricci and scalar curvatures of the associated pseudo-Riemannian metric. This leads naturally to a discussion of the Einstein condition in this setting.

In the final section of this chapter we consider Künneth structures compatible with a positive definite Kähler metric, and we show that in this case the Künneth structure and the Kähler metric are flat.

In this chapter we discuss the unique torsion-free affine connection defined by a Künneth structure. This connection, which we call the Künneth connection, preserves the two foliations and is compatible with the symplectic structure. We will actually start with a more general setup, proving that for every almost Künneth structure there is a distinguished connection for which the whole structure is parallel. It then turns out that this connection is torsion-free if and only if the almost Künneth structure is integrable, i.e. it arises tautologically from a Künneth structure. In this case the Künneth connection is just the Levi-Civita connection of the associated pseudo-Riemannian metric.

In the final section of this chapter we use connections to prove that Künneth or bi-Lagrangian structures are in fact the same as para-Kähler structures.

This book is about the geometry and topology of symplectic manifolds equipped with pairs of complementary Lagrangian foliations. The resulting structure is very rich, intertwining symplectic geometry, the theory of foliations, dynamical systems, and pseudo-Riemannian geometry in interesting ways.

Before describing the contents of the book in detail, we want to discuss a few motivating vignettes. The first two of these are to be kept in mind as motivational background, whereas the third and fourth ones will be taken up again and again later in the book.

With this chapter we begin the main part of our discussion of Künneth geometry. We first define Künneth and almost Künneth structures. The former are the main structures whose geometry we investigate in this and the coming chapters. In this chapter we give only their most basic properties, and we discuss their automorphism groups. Most of this chapter consists of the discussion of several classes of examples.

Whenever one is given two non-degenerate 2-forms ω and η on the same manifold M, there exists a unique field of invertible endomorphisms A of the tangent bundle TM defined by the equation iXω = iAXη. The important special case when the two 2-forms involved are closed, and therefore symplectic, is very interesting both from the point of view of physics, where it arises in the context of bi-Hamiltonian systems, and from a purely mathematical viewpoint. In physics the field of endomorphisms A is called a recursion operator, and we adopt this terminology here.

In Section 8.1 we consider the simplest examples, where the recursion operator A satisfies A2 = ±1. We find that these most basic cases correspond precisely to symplectic pairs and to holomorphic symplectic forms respectively. In Section 8.2 we formulate the basics of hypersymplectic geometry in the language of recursion operators. The definition we give is not the original one due to Hitchin [Hit-90], but is equivalent to it. In Section 8.3 we show that every hypersymplectic structure contains a family of Künneth structures parametrised by the circle. The associated metric is independent of the parameter, and is Ricci-flat, cf. Section 8.4.

This chapter is a crash course on symplectic geometry. We first introduce symplectic manifolds and their Lagrangian submanifolds and give some examples. Then we discuss the Moser method [Mos-65], and use it to prove results about the existence of local normal forms and of standard tubular neighbourhoods. This is of course just the beginning of the vast area of symplectic geometry. A more extensive account is contained, for example, in the book of McDuff-Salamon [McS-95]; see especially Chapter 3 of that book. Other useful references for this material include the writings of Weinstein [Wei-71, Wei-77].