Sasakian structures on CR-manifolds

By L. Ornea, Misha Verbitsky. 2006

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A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of $$\mathbb{R}$$, with the symplectic form homogeneous of degree 2. If X is also Kähler, and its metric is homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler 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 S1-bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding of M into 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.

View Paper: https://www.semanticscholar.org/paper/34879ae5bce9f4c1aef806817bb920a8489d4378