Affine connections on homogeneous hypercomplex manifolds
The paper deals with hypercomplex structures and related connections. The basic definitions and results on almost hypercomplex structures are recalled. An almost hypercomplex structure on a manifold \(M=M^{2n}\) is a triple \(H=(J_1,J_2,J_3)\) of anticommuting almost complex structures with \(J_3=J_1J_2\); the Obata connection of \(H=(J_\alpha)\) is the (uniquely defined) affine connection \(\nabla^H\) on \(M\) which preserves \(H\), i.e. \(\nabla^HJ_\alpha=0, \alpha =1,2,3\), and with torsion \(T^H=\frac{1}{6}\sum_\alpha N_\alpha\), where \(N_\alpha\) is the Nijenhuis tensor of \(J_\alpha, \alpha=1,2,3\). The condition of preserving \(H\) is equivalent to \(\text{Hol}(\nabla^H) \subset GL(n,\mathbb{H})\), where \(\text{Hol}(\nabla^H)\) is the holonomy group of \(\nabla^H\). \(H=(J_\alpha)\) is called a hypercomplex structure if the almost complex structures \(J_\alpha\) are integrable, i.e., \(N_\alpha =0\) on \(M\), or equivalently, the Obata connection is torsionless. A hypercomplex structure \(H\) is called integrable if the \(GL(n,\mathbb{H})\)-structure of \(M\) which is naturally associated to it is integrable. M. Obata proved that \(H\) is integrable if and only if \(\nabla^H\) is flat. The author studies in particular the homogeneous hypercomplex manifolds \((M,H)\), i.e., the homogeneous manifolds \(M=G/K\) where a \(G\)-invariant hypercomplex structure \(H\) is given. For this purpose, for any G-invariant almost hypercomplex structure \(H\) on \(M=G/K\), she introduces an affine connection which is closely related to the Obata connection and whose torsion vanishes if and only if \(H\) is an hypercomplex structure of a certain special type (an abelian hypercomplex structure). A classification of the 4-dimensional real Lie groups which admit an invariant almost hypercomplex structure is given and the hypercomplex structures which are integrable are determined. As a consequence, a self-dual, non-flat, Ricci flat affine connection on \({\mathbb{R}}^4\) admitting a solvable simply transitive group of affine transformations is obtained. This result is extended to a family of hypercomplex manifolds of dimension \(4n, n>1\), considered in [\textit{M. L. Barberis} and \textit{I. Dotti Miatello}, Q. J. Math. Oxf., II. Ser. 47, 389-404 (1996; Zbl 0874.53023)]. A sufficient condition for the integrability of hypercomplex structures on certain solvable Lie algebras is also given.
scientific article; zbMATH DE number 1369696
scientific article