Kronheimer’s work in this period was mainly concerned with the subject of “hyperkahler geometry”. Recall that a Kahler manifold is a Riemannian manifold with a complex structure I on its tangent spaces which is preserved by the parallel transport of the Levi-Civita connection: equivalently, it is a manifold whose holonomy group reduces to the unitary group. A hyperkahler manifold is a Riemannian manifold with a triple of complex structures I, J, K, satisfying the algebraic identities of the quaternions, each of which yields a Kahler structure. Equivalently, it is a manifold whose holonomy group reduces to the compact symplectic group. A hyperkahler structure gives a natural kind of “quaternionic manifold” and they are also very interesting from the point of view of Riemannian geometry: in any dimension they have vanishing Ricci tensor, and in dimension 4 they can be characterised, locally, as metrics which are simultaneously Ricci-flat and “self-dual”.
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