On some Moment Maps and Induced Hopf Bundles in the Quaternionic Projective Space

We describe a diagram containing the zero sets of the moment maps associated to the diagonal U(1) and Sp(1) actions on the quaternionic projective space HP^n. These sets are related both to focal sets of submanifolds and to Sasakian-Einstein structures on induced Hopf bundles. As an application, we construct a complex structure on the Stiefel manifolds V_2 (C^{n+1}) and V_4 (R^{n+1}), the one on the former manifold not being compatible with its known hypercomplex structure.Comment: Revised version, a more complete proof of a statement and some references were added. LaTex, 21 pages, to be published in Int. J. Mat

Complex Structures on some Stiefel Manifolds

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).Comment: LaTex, 11 pages, to be published in Bull. Soc. Sc. Math. Roumanie, Volume in memory of G. Vrancean

Sasakian structures on CR-manifolds

A contact manifold $M$ can be defined as a quotient of a symplectic manifold $X$ by a proper, free action of $\R^{>0}$, with the symplectic form homogeneous of degree 2. If $X$ is, in addition, Kaehler, and its metric is also homogeneous of degree 2, $M$ is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kaehler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold $M$ is CR-diffeomorphic to an $S^1$-bundle of unit vectors in a positive line bundle on a projective K\"ahler orbifold. This induces an embedding from $M$ to an algebraic cone $C$. We show that this embedding is uniquely defined by the CR-structure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.Comment: 23 pages, v. 1.1: replaced the abstract, no change in the paper itsel