Hypercomplex structures on Stiefel manifolds. (English) Zbl 0843.53030

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 \(h({\mathbf p})\).


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
32Q99 Complex manifolds
53C56 Other complex differential geometry
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
32M05 Complex Lie groups, group actions on complex spaces
Full Text: DOI


[1] Battaglia, F.: A hypercomplex Stiefel manifold. Preprint 1993. · Zbl 0860.53046
[2] Beauville, A.: Varieétés Kählèriennes dont la lère classe de Chern. J. Differ. Geom. 18 (1983), 755-782. · Zbl 0537.53056
[3] Besse, A.L.: Einstein manifolds. Springer-Verlag, New York 1987. · Zbl 0613.53001
[4] Bonan, E.: Sur les G-structures de type qua ternionien. Cah. Topologie Géom. Différ. Catégoriques 9 (1967), 389-461.
[5] Boyer, C.P.: A note on hyperhermitian four-manifolds. Proc. Am. Math. Soc. 102 (1988), 157-164. · Zbl 0642.53073
[6] Boyer, C.P.; Galicki, K.; Mann, B.M.: Quaternionic reduction and Einstein manifolds. Commun. Anal. Geom. 1 (1993) 2, 229-279. · Zbl 0856.53038
[7] Boyer, C.P.; Galicki, K.; Mann, B.M.: The geometry and topology of 3-Sasakian Manifolds. J. Reine Angew. Math. 455 (1994), 183-220. · Zbl 0889.53029
[8] Boyer, C.P.; Galicki, K.; Mann, B.M.: Some New Examples of Compact Inhomogeneous Hypercomplex Manifolds. Math. Res. Lett. 1 (1994), 531-538. · Zbl 0841.53041
[9] Boyer, C.P.; Galicki, K.; Mann, B.M.: Hypercomplex Structures on Circle Bundles. In Preparation. · Zbl 0843.53030
[10] Galicki, K.; Lawson, Jr., B.H.: Quaternionic reduction and quaternionic orbifolds. Math. Ann. 282 (1988), 1-21. · Zbl 0628.53060
[11] Griffiths, P.A.: Some geometric and analytic properties of homogeneous complex manifolds, parts I and III. Acta. Math. 110 (1963), 115-208. · Zbl 0171.44601
[12] Hernandez, G.: On hyper f-structures. Dissertation, Univ. of New Mexico 1994.
[13] Joyce, D.: Compact hypercomplex and quaternionic manifolds. J. Differ. Geom. 35 (1992) , 743-762. · Zbl 0735.53050
[14] Joyce, D.: The hypercomplex quotient an the quaternionic quotient. Math. Ann. 290 (1991), 323-340. · Zbl 0723.53043
[15] Molino, P.: Riemannian Foliations. Birkhäuser, Boston 1988.
[16] Obata, M.: Affine connections on manifolds with almost complex, quaternionic or Hermitian structure. Jap. J. Math. 26 (1955), 43-79. · Zbl 0089.17203
[17] Salamon, S.: Differential geometry of quaternionicmanifolds. Ann. Sci. Éc. Norm. Supér. 19 (1986), 31-55. · Zbl 0616.53023
[18] Samelson, H.: A class of complex analytic manifolds. Port. Math. 12 (1953), 129-132. · Zbl 0052.02405
[19] Spindel, Ph.; Sevrin, A.; VanProeyen, A.: Extended supersymmetric ?-models on group manifolds. Nucl. Phys. B 308 (1988), 662-698.
[20] Spivak, M.: A Comprehensive Introduction to Differential Geometry. Vol. IV, Publish or Perish, Inc., 1975. · Zbl 0306.53001
[21] Wang, H.C.: Closed manifolds with homogeneous complex structures. Am. J. Math. 76 (1954), 1-32. · Zbl 0055.16603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.