HYPER-KÄHLER METRICS CONFORMAL TO LEFT INVARIANT METRICS ON FOUR DIMENSIONAL LIE GROUPS

M.L. BARBERIS

MATHEMATICAL PHYSICS, ANALYSIS AND GEOMETRY

SPRINGER

Lugar: Dordrecht; Año: 2003 vol. 6 p. 1 - 8

1385-0172

Let g be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold (M;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 explicitely the corresponding hyper-Kähler metrics, which are of cohomegeneity 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.