Reading about Sasakian manifolds one come across two slogans:


A) "A Sasakian manifold is an odd-dimensional analogue of a Kahler manifold."


B) "A Sasakian manifold sits between two Kahler manifolds - one above and one below."


I would like to understand the second slogan for the motivating example 
of the three sphere $S^3$. What are the two Kahler manifolds that it sits 
between? I would guess that below is the projective line $\mathbb{CP}^1$. But 
I cannot guess what lies above. 



$S^3$ as a Sasakian Manifold
I will answer your question for $S^{2n+1}$, since there is no difference between the case $n=1$ and the case of general $n$.


Let $(M^{2n+1},g,\theta)$ be a Sasakian manifold. One definition of a Sasakian manifold is that its metric cone is Kähler; this is the "one above". Here the *metric cone* is the manifold $M\times(0,\infty)$ with metric $dt^2+t^2g$. Thus in the case of $S^{2n+1}$, the metric cone is $\mathbb{C}^{n+1}\setminus\{0\}$ with the flat metric (written in spherical coordinates).


Taking the quotient by the $S^1$-action on $(M^{2n+1},\theta)$ determined by the Reeb vector field gives the "Kähler manifold below". For $S^{2n+1}$, the $S^1$-action is scalar multiplication by $e^{i\phi}$ (regarding $S^{2n+1}$ as the unit sphere in $\mathbb{C}^{n+1}$), so the quotient is $\mathbb{C}P^n$.


