Affine Connections on Homogeneous Hypercomplex Manifolds

M.L. BARBERIS

JOURNAL OF GEOMETRY AND PHYSICS

ELSEVIER SCIENCE BV

Año: 1999 vol. 32 p. 1 - 13

0393-0440

It is the aim of this work to study affine connections whose holonomy group is contained in Gl(n;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 [Obata, Japan. J. Math. 1956]. 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 explicitely 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 at affine connection on R4 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 [Barberis-Dotti Miatello, Quart. J. Math Oxford 1996]. We also give a sufficient condition for the integrability of hypercomplex structures on certain solvable Lie algebras.