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.