Homogeneous hyper-Hermitian metrics which are conformally hyper-Kähler

Details Homogeneous hyper-Hermitian metrics which are conformally hyper-Kähler Homogeneous hyper-Hermitian metrics which are conformally hyper-Kähler

2000-09-04

arxiv.org/abs/math/0009035v1

Math. Phys. Anal. Geom. 6 (2003), 1--8

Mathematics

Differential Geometry

6 pages

Scientific paper

Let $g$ be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold $M$. We show that when the isometry group $I(M,g)$ contains a subgroup acting simply transitively on $M$ by hypercomplex isometries then the metric $g$ is conformal to a hyper-K\"ahler metric. We describe explicitely the corresponding hyper-K\"ahler metrics and it follows that, in four dimensions, these are the only hyper-K\"ahler metrics containing a homogeneous metric in its conformal class.

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