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  • 7 - The Curvature of a Künneth Structure

  • M. J. D. Hamilton, Universität Stuttgart, D. Kotschick, Ludwig-Maximilians-Universität München
  • Book:
    Künneth Geometry
    Published online:
    07 December 2023
    Print publication:
    21 December 2023, pp 85-106

  • Summary

    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.