Two Aspects of Sasakian Geometry
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Sasakian structures are the counterparts of Kahler structures in odd dimensions. A Sasakian manifold is a strictly pseudo-convex CR manifold with a Reeb vector field that generates CR automorphisms. We first study Type I deformations of Sasakian structures, which amount to different choices of Reeb vector field on a fixed CR manifold. Here we show that the CR automorphism group of a Sasakian manifold is severely constrained by mild curvature assumptions. We then study products of pairs of compact Sasakian manifolds. Such products are shown to always yield compact complex manifolds that do not admit Kahler metrics, generalizing a remarkable construction due to Calabi and Eckmann. As a consequence, any product of two compact Sasaki-Einstein manifolds yields an Einstein Hermitian metric on a compact complex manifold which does not admit Kahler metrics. This result stands in marked contrast to the situation in real dimension 4, where LeBrun showed that Einstein Hermitian metrics on compact complex surfaces are always conformally Kahler.