On 4-dimensional CR-submanifolds of a 6-dimensional sphere. (English) Zbl 1029.53066
Fukaya, Kenji (ed.) et al., Minimal surfaces, geometric analysis and symplectic geometry. Based on the lectures of the workshop and conference, Johns Hopkins University, Baltimore, MD, USA, March 16-21, 1999. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 34, 143-154 (2002).
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].
For the entire collection see [Zbl 0994.00028].
Reviewer: Mircea Crâşmăreanu (Iaşi)
MSC:
53C40 | Global submanifolds |
57R42 | Immersions in differential topology |
57R20 | Characteristic classes and numbers in differential topology |