The aim of this paper is to give some topological restrictions on the existence of compact 4-dimensional CR-submanifolds of \(S^{6}.\) By using Hopf’s index theorem and some own suitable results and remarks, the author obtains new results. If \(M^{4}\) is an oriented 4-dimensional CR-submanifold of \(S^{6}\) then: a) both the Euler class of \(M^{4}\) and the Euler class of the complex sub-bundle \(H\) over \(M\) vanish, b) the first Pontrjagin class of \(M^{4}\) vanishes. In particular, if \(M^{4}\) is compact, its Hirzebruch signature is equal to zero and its Euler number \(\chi (M^{4})\) is also equal to zero. It is proved that \(S^{4},\) \(S^{2}\times S^{2}\) and \(\mathbb{C}\mathbb{P}^{2}\) can not be immersed into \(S^{6}\) as a CR-submanifold. Besides proving several fundamental properties of 4-dimensional CR-submanifolds of a nearly Kähler 6-dimensional sphere, the authors construct explicit examples of such submanifolds.
For the entire collection see [Zbl 0994.00028].