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.