Abstract
This paper describes a family of hypercomplex structures {% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFqessaaa!4076!\[\mathcal{I}\]a(p)}a=1,2,3 depending on n real non-zero parameters p = (p 1,...,p n) on the Stiefel manifold of complex 2-planes in ℂn for all n > 2. Generally, these hypercomplex structures are inhomogenous with the exception of the case when all the p i's are equal. We also determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure. Furthermore, we solve the equivalence problem for the hypercomplex structures in the case that the components of p are pairwise commensurable. Finally, some of these examples admit discrete hypercomplex quotients whose topology we also analyze.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Battaglia, F.: A hypercomplex Stiefel manifold. Preprint 1993.
Beauville, A.: Varieétés Kählèriennes dont la lère classe de Chern. J. Differ. Geom. 18 (1983), 755–782.
Besse, A.L.: Einstein manifolds. Springer-Verlag, New York 1987.
Bonan, E.: Sur les G-structures de type qua ternionien. Cah. Topologie Géom. Différ. Catégoriques 9 (1967), 389–461.
Boyer, C.P.: A note on hyperhermitian four-manifolds. Proc. Am. Math. Soc. 102 (1988), 157–164.
Boyer, C.P.; Galicki, K.; Mann, B.M.: Quaternionic reduction and Einstein manifolds. Commun. Anal. Geom. 1 (1993) 2, 229–279.
Boyer, C.P.; Galicki, K.; Mann, B.M.: The geometry and topology of 3-Sasakian Manifolds. J. Reine Angew. Math. 455 (1994), 183–220.
Boyer, C.P.; Galicki, K.; Mann, B.M.: Some New Examples of Compact Inhomogeneous Hypercomplex Manifolds. Math. Res. Lett. 1 (1994), 531–538.
Boyer, C.P.; Galicki, K.; Mann, B.M.: Hypercomplex Structures on Circle Bundles. In Preparation.
Galicki, K.; Lawson, Jr., B.H.: Quaternionic reduction and quaternionic orbifolds. Math. Ann. 282 (1988), 1–21.
Griffiths, P.A.: Some geometric and analytic properties of homogeneous complex manifolds, parts I and III. Acta. Math. 110 (1963), 115–208.
Hernandez, G.: On hyper f-structures. Dissertation, Univ. of New Mexico 1994.
Joyce, D.: Compact hypercomplex and quaternionic manifolds. J. Differ. Geom. 35 (1992) , 743–762.
Joyce, D.: The hypercomplex quotient an the quaternionic quotient. Math. Ann. 290 (1991), 323–340.
Molino, P.: Riemannian Foliations. Birkhäuser, Boston 1988.
Obata, M.: Affine connections on manifolds with almost complex, quaternionic or Hermitian structure. Jap. J. Math. 26 (1955), 43–79.
Salamon, S.: Differential geometry of quaternionicmanifolds. Ann. Sci. Éc. Norm. Supér. 19 (1986), 31–55.
Samelson, H.: A class of complex analytic manifolds. Port. Math. 12 (1953), 129–132.
Spindel, Ph.; Sevrin, A.; VanProeyen, A.: Extended supersymmetric σ-models on group manifolds. Nucl. Phys. B 308 (1988), 662–698.
Spivak, M.: A Comprehensive Introduction to Differential Geometry. Vol. IV, Publish or Perish, Inc., 1975.
Wang, H.C.: Closed manifolds with homogeneous complex structures. Am. J. Math. 76 (1954), 1–32.
Author information
Authors and Affiliations
Additional information
During the preparation of this work all three authors were supported by NSF grants.
Rights and permissions
About this article
Cite this article
Boyer, C.P., Galicki, K. & Mann, B.M. Hypercomplex structures on Stiefel manifolds. Ann Glob Anal Geom 14, 81–105 (1996). https://doi.org/10.1007/BF00128197
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00128197