Normal CR structures on \(S^3\)
Let \((M,\alpha)\) be a contact manifold of dimension \(2n+1\). \(\alpha\) determines a vector field \(R\), called Reeb vector field, and a contact distribution \(\xi=\text{ker }\alpha\). Given a complex structure \(J:\xi\to \xi\) compatible with \(d\alpha_\xi\) the pair \((J,\xi)\) is called a normal CR structure on \(M\) if \(L_{R\xi }J\equiv 0\) for the Lie derivative. In this paper the author gives the classifications of the normal CR structures on \(S^3\) and their automorphism groups. Combing with results in another paper [Math. Z. 238, No. 3, 441--460 (2001; Zbl 1043.32020)] this completes the classifications of the normal CR structures on compact \(3\)-manifolds. Moreover the author gives a criterion to compare a structure to normal CR structures and shows that the underlying contact structure is unique up to diffeomorphism.
scientific article; zbMATH DE number 1944235