Affine connections on homogeneous hypercomplex manifolds

It is the aim of this work to study affine connections whose holonomy group is contained in Cl(nt H). These connections arise in the context of hypercomplex geometry. We study the case of homogeneous hypercomplex manifolds and introduce an affine connection which is closely related to the Obata connection [M. Obata, Japan J. Math. 26 (1956) 43-77]. We find a family of homogeneous hypercomplex manifolds whose corresponding connections are not flat with holonomy contained in Sl(n, H). We consider first the 4-dimensional case and determine all the 4-dimensional real Lie groups which admit integrable invariant hypercomplex structures. We describe explicitly the Obata connection corresponding to these structures and by studying the vanishing of the curvature tensor, we determine which structures are integrable, obtaining as a byproduct a self-dual, non-flat, Ricci flat affine connection on R-4 admitting a simply transitive solvable group of affine transformations. This result extends to a family of hypercomplex manifolds of dimension 4n, n > 1, considered in [M.L. Barberis, LD. Miatello, Quart. J. Math. Oxford 47 (2) (1996) 389-404]. We also give a sufficient condition for the integrability of hypercomplex structures on certain solvable Lie algebras. (C) 1999 Elsevier Science B.A. All right reserved.