Hyper-Kähler metrics conformal to left invariant metrics on four-dimensional Lie groups. (English) Zbl 1031.53074

Let \(g\) be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold \((M,{\mathcal H})\). We show that when the isometry group \(I(M,g)\) contains a subgroup \(G\) acting simply transitively on \(M\) by hypercomplex isometries, then the metric \(g\) is conformal to a hyper-Kähler metric. We describe explicitly the corresponding hyper-Kähler metrics, which are of cohomogeneity one with respect to a 3-dimensional normal subgroup of \(G\). It follows that, in four dimensions, these are the only hyper-Kähler metrics containing a homogeneous metric in its conformal class.
Reviewer: D.Perrone (Lecce)


53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C30 Differential geometry of homogeneous manifolds
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