By Frederick S. And Frederick H. Bailey Woods

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**Example text**

They are Riemannian manifolds (M A n , g) with locally defined almost complex structures J1, J2, J3, which span a globally defined rank-3 bundle Q c End(TM), such that Q is preserved by the Levi-Civita connection. Equivalently one can define quaternionic Kähler manifolds as Riemannian manifolds with holonomy Sp(rc) -Sp(l). (Some modification is usually taken ([Co], p. ) It is not difficult to show that quaternionic Kähler manifolds are Einstein and in particular their scalar curvature is constant.

For more information see Section 4 in [Bel]. - Kühler's zeta function a zeta function associated to certain subvarieties of arithmetic varieties going back to the last chapter of [28]. For a definition see [Be2]. ), or, as a variant, one has a whole 2-sphere of Kähler structures, - quaternionic Kähler manifold a Riemannian manifold (A/ 4 ", g) with locally defined almost complex structures J\, Ji, h, which span a globally defined rank-3 bundle Q C E n d ( T M ) , such that Q is preserved by the Levi-Civita connection.

Can be extended by D to a new Z2-graded Lie algebra 0 = 0(Π) : = MD + po + Ρι· The adjoint representation of 0 is faithful and hence defines on g the structure of a linear Lie algebra. LetG = G(FI) c Aut g denote the corresponding connected linear group. Moreover, Κ = Κ (Ε) c G is the connected linear Lie group with Lie algebra t = o(E) 0 oiE^-), where Ε c V is a three-dimensional Euclidean subspace. Cortes has as his (first) principal result ([Co], Theorem A, p. 4, resp. Theorem 8, p. ) Μ = G/K has G-equivariant quaternionic structures.