Sasakian structures on CR-manifolds

Details Sasakian structures on CR-manifolds Sasakian structures on CR-manifolds

2006-06-06

arxiv.org/abs/math/0606136v2

Geom. Dedicata 125 (2007), 159--173.

Mathematics

Differential Geometry

23 pages, v. 1.1: replaced the abstract, no change in the paper itself

Scientific paper

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.

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