Hypercomplex structures on Stiefel manifolds
This paper proves the existence of uncountably many distinct hypercomplex structures \(\{{\mathcal I}^a ({\mathbf p})\}\) on Stiefel manifolds \(V\) of complex 2-planes in complex \(n\)-space, and studies their properties. A hypercomplex structure on a smooth manifold \(M\) is a \(\text{GL} (n, \mathbb{H})\)-structure preserved by a torsion-free connection. In particular every such \(M\) has three complex structures \(I\), \(J\) and \(K\) satisfying algebraic-relations of imaginary quaternions. If \(n>2\) and \({\mathbf p}= (p_1, p_2, \dots, p_n)\in (\mathbb{R}^*)^n\), the first result asserts that for each such \({\mathbf p}\) there is a compact hypercomplex manifold \(({\mathcal N} ({\mathbf p}), {\mathcal I} ({\mathbf p}))\), where \({\mathcal N} ({\mathbf p})\) is diffeomorphic to the Stiefel manifold \(V\). To study these hypercomplex structures let \(C_n= \{{\mathbf p}\in \mathbb{R}^n\mid 0< p_1\leq p_2\leq \dots \leq p_n\}\) be the positive cone and \({\mathbf p}\) be commensurable if each of the ratios \(p_i/ p_j\) is a rational number. From the property that permuting the coordinates of \({\mathbf p}\) or changing the signs does not change the hypercomplex structure represented by \({\mathbf p}\), one assumes that \({\mathbf p}\in C_n\). The second result asserts that if \({\mathbf p}\) and \({\mathbf q}\) are both commensurable sequences in \(C_n\), then \({\mathcal N} ({\mathbf p})\) and \({\mathcal N} ({\mathbf q})\) are hypercomplex equivalent if and only if \({\mathbf p}= {\mathbf q}\). Furthermore \({\mathcal N} ({\mathbf p})\) is hypercomplex homogeneous if and only if \({\mathbf p}= \lambda (1, 1, \dots, 1)\) for some \(\lambda\in \mathbb{R}^*\). The discrete quotients are also studied. A commensurable \({\mathbf p}\) is called basic if all \(p_i\) are integers and the greatest common divisor of all \(p_i\) is one. A basic \({\mathbf p}\) is called coprime if the \(p_i\) are pairwise relatively prime, and \(k\)-coprime if \({\mathbf p}\) is an integer multiple of a basic sequence such that \((p_i, p_j, k)\) have no common factor for \(1\leq i< j\leq n\). Let \({\mathbf p}\) be a \(k\)-coprime. Then there is a compact hypercomplex manifold \({\mathcal H} ({\mathbf p}, k)\) with universal cover \(\rho_k: {\mathcal N} ({\mathbf p})\to {\mathcal H} ({\mathbf p}, k)\) such that \(\pi_1 ({\mathcal H} ({\mathbf p}, k))\cong \mathbb{Z}_k\) and \(\rho_k\) is a hypercomplex map. Finally, this paper determines the connected component of the group of hypercomplex automorphisms of \({\mathcal N} ({\mathbf p})\), and shows the existence of a natural hyperhermitian metric \(h({\mathbf p})\) such that every infinitesimal automorphism is an infinitesimal isometry with respect to \(h({\mathbf p})\).
scientific article; zbMATH DE number 854360