Einstein four-manifolds with skew torsion

We develop a notion of Einstein manifolds with skew torsion on compact, orientable Riemannian manifolds of dimension four. We prove an analogue of the Hitchin–Thorpe inequality and study the case of equality. We use the link with self-duality to study the moduli space of 1-instantons on S4S4 for a family of metrics defined by Bonneau.

► We propose a notion for Einstein four-manifolds with skew torsion.
► We show an analogue of the Hitchin–Thorpe inequality.
► We study the case of equality and show that it only occurs for manifolds of the type S1×S3S1×S3.
► We prove that in the compact case, when the torsion is closed, we have two instantons on Λ+Λ+.
► We show that these two instantons are not always gauge equivalent.