A Sasakian manifold is often said to be the **odd dimensional analogue** of a Kähler manifold. 


Now for a $2n$-dimensional Kähler manifold we know from [Atiyah](https://mathoverflow.net/questions/262213/which-kahler-manifolds-are-spin) that it is spin exactly if the line bundle $\Omega^{(0,n)}$ admits a square root ${\cal S}$, and a choice of spin structure is equivalent to a choice of holomorphic structure ${\cal S}$. In this case the associated spinor bundle is the tensor product of ${\cal S}$ with the anti-holomorphic complex. Moreover, the associated Dirac operator is the tensor product of $\overline{\partial} + \overline{\partial}^\*$ with the $\overline{\partial}$-operator corresponding to the choice of holomorphic structure. 


So does the above analogy present any Sasakian versions of these spin geometry to complex geometry dictionary? 



Spin structures on Sasakian manifolds and the Kähler analogy
Every Sasakian manifold $M$ (of dimension $2k+1$) has a canonical $\mathrm{Spin}^c$ structure, because the cone $\overline{M}$ over $M$ is Kähler and thus has a canonical $\mathrm{Spin}^c$ structure which restricts to $\mathrm{Spin}^c$ structure on $M$. 


If $M$ is Einstein, then the cone $\overline{M}$ is Ricci flat and that in turn implies that the auxiliary bundle $\Lambda^{k+1, 0} \overline{M}$ of the canonical $\mathrm{Spin}^c$ structure on the cone $\overline{M}$ is flat. Now $\mathrm{Spin}^c$ structures on simply connected manifolds with trivial auxiliary bundle is canonically identified with a spin strucutre.


For more details see *Moroianu, Andrei*: [Parallel and Killing spinors on $Spin^c$ manifolds](http://www.ams.org/mathscinet-getitem?mr=1463835), Comm. Math. Phys. 187 (1997), no. 2, 417–427


