CR-structures on $S^3$

Question

How can I understand the hypersurface $CR$-structure on $S^3$? And what $CR$-structure makes $S^3$ a Levi-flat $CR$-manifold?

Answer 1

The standard CR-structure on $S^3$ comes from the embedding into the complex manifold $\mathbb C^2=\mathbb R^4$ as a hypersurface. Explicitly, let $\langle\ ,\ \rangle$ be the standard Hermitian inner product on $\mathbb C^2$, so $S^3=\{z\in\mathbb C^2:\langle z,z\rangle=1\}$. This shows that for $z\in S^3$ you get $T_zS^3=\{w\in\mathbb C^2:Re(\langle z,w\rangle)=0\}$, and the CR-subspsace is the maximal complex subspace in there, i.e. $H_z=\{w\in\mathbb C^2:\langle z,w\rangle=0\}$.

I don't know whether $S^3$ admits a Levi-flat CR structure.