A smooth foliation of the 5-sphere by complex surfaces (Q1860983)

The authors construct an example of a smooth codimension one foliation \(\mathcal F\) on the sphere \({\mathbf S}^5\) whose tangent \(T{\mathcal F}\) has an integrable almost complex structure; that is, they construct a smooth, codimension one, integrable and Levi flat CR-structure on \({\mathbf S}^5\). This CR-structure is not embeddable in any Stein space nor Kähler manifold. The construction is a smart modification of the first example of a smooth codimension one foliation of \({\mathbf S}^5\), discovered by \textit{H. B. Lawson} [Ann. Math. (2) 94, 494-503 (1971; Zbl 0236.57014)].