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.