Maria Laura Barberis

On certain locally homogeneous Clifford manifolds

Given a manifoldM, a Clifford structure of orderm onM is a family ofm anticommuting complex structures generating a subalgebra of dimension 2m of End(T(M)). In this paper we investigate the existence of locally invariant Clifford structures of orderm≥2 on a class of locally homogeneous manifolds. We study the case of solvable extensions ofH-type groups, showing in particular that the solvable Lie groups corresponding to the symmetric spaces of negative curvature carry invariant Clifford structures of orderm≥2. We also show that for eachm and any finite groupF, there is a compact flat manifold with holonomy groupF and carrying a Clifford structure of orderm.

Affine connections on homogeneous hypercomplex manifolds

Abstract 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 [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, I.D. 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.

Hyper-Kähler Metrics Conformal to Left Invariant Metrics on Four-Dimensional Lie Groups

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.

COMPLEX CONNECTIONS WITH TRIVIAL HOLONOMY

Given an almost complex manifold (M, J), we study complex connections with trivial holonomy such that the corresponding torsion is either of type (2, 0) or of type (1, 1) with respect to J. Such connections arise naturally when considering Lie groups, and quotients by discrete subgroups, equipped with bi-invariant and abelian complex structures.

Conformal Killing 2-forms on four-dimensional manifolds

We study four-dimensional simply connected Lie groups G with a left invariant Riemannian metric g admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action, or the metric is half-conformally flat. In the first case, the problem reduces to the study of invariant conformally Kähler structures, whereas in the second case, the Lie algebra of G belongs (up to homothety) to a finite list of families of metric Lie algebras.