Complex Structures on some Stiefel Manifolds

Abstract

We discuss conditions for the integrability of an almost complex structure defined on the total space of an induced Hopf S^3-bundle over a Sasakian manifold . As an application, we obtain an uncountable family of inequivalent complex structures on the Stiefel manifolds of orthonormal 2-frames in C^{n+1}, non compatible with its standard hypercomplex structure. Similar families of complex structures are constructed on the Stiefel manifold of oriented orthonormal 4-frames in R^{n+1}, as well as on some special Stiefel manifolds related to the groups G_2 and Spin(7).