Hypercomplex structures on Stiefel manifolds
type: Item
instance of: scholarly article
title: Hypercomplex structures on Stiefel manifolds
zbMATH Open document ID: 0843.53030
DOI: 10.1007/BF00128197
author: Krzysztof Galicki
author: Charles P. Boyer
author: Benjamin M. Mann
published in: Annals of Global Analysis and Geometry
publication date: 1996-04-29T00:00:00Z
review text: 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 integ…
reviewed by: I-Hsun Tsai
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 32Q99
Mathematics Subject Classification ID: 53C56
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 32M05
zbMATH DE Number: 854360
zbMATH Keywords: foliations
zbMATH Keywords: existence of hypercomplex structures
zbMATH Keywords: Stiefel manifolds
zbMATH Keywords: hypercomplex automorphism
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 854360