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.

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.