# Three-Dimensional Minimal CR Submanifolds of the Sphere *S* ^{ 6 } **(1)** Contained in a Hyperplane

- 127 Downloads
- 5 Citations

## Abstract

It is well known that the sphere *S* ^{6}(1) admits an almost complex structure *J*, constructed using the Cayley algebra, which is nearly Kähler. Let *M* be a Riemannian submanifold of a manifold \({\widetilde{M}}\) with an almost complex structure *J*. It is called a CR submanifold in the sense of Bejancu (Geometry of CR Submanifolds. D. Reidel Publ. Dordrecht, 1986) if there exists a *C* ^{∞}-differentiable holomorphic distribution \({\mathcal D_1}\) in the tangent bundle such that its orthogonal complement \({\mathcal D_2}\) in the tangent bundle is totally real. If the second fundamental form vanishes on \({\mathcal D_i}\), the submanifold is \({\mathcal D_i}\)-geodesic. The first example of a three-dimensional CR submanifold was constructed by Sekigawa (Tensor N S 41:13–20, 1984). This example was later generalized by Hashimoto and Mashimo (Nagoya Math J 156:171–185, 1999). Note that both the original example as well as its generalizations are \({\mathcal D_1}\)- and \({\mathcal D_2}\)-geodesic. Here, we investigate the class of the three-dimensional minimal CR submanifolds *M* of the nearly Kähler sphere *S* ^{6}(1) which are not linearly full. We show that this class coincides with the class of \({\mathcal D_1}\)- and \({\mathcal D_2}\)- geodesic CR submanifolds and we obtain a complete classification of such submanifolds. In particular, we show that apart from one special example, the examples of Hashimoto and Mashimo are the only \({\mathcal D_1}\)- and \({\mathcal D_2}\)-geodesic CR submanifolds.

## Keywords

CR submanifold Minimal submanifold nearly Kähler six-sphere linearly full \({\mathcal D}\)-geodesic submanifolds## Mathematics Subject Classification

Primary 53B20 Secondary 53B21 53B25## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Antić M.: 4-dimensional minimal CR submanifolds of the sphere
*S*^{6}contained in a totally geodesic sphere*S*^{5}. J. Geom. Phys.**60**, 96–110 (2010)MATHMathSciNetCrossRefGoogle Scholar - 2.Bejancu, A.: Geometry of CR-Submanifolds. D. Reidel Publ., Dordrecht (1986)Google Scholar
- 3.Calabi, E., Gluck, H.: What are the best almost complex structures on the 6-sphere. In: Greene, R.E., Yau, S.S.-T. (eds.) Differential Geometry: Geometry in Mathematical Physics and Related Topics, pp. 99–106. American Mathematical Society, Providence (1993)Google Scholar
- 4.Chen B.Y.: Some pinching and classification theorems for minimal submanifolds. Archiv. Math. (Basel)
**60**, 568–578 (1993)MATHCrossRefGoogle Scholar - 5.Djorić M., Vrancken L.: Three dimensional minimal CR submanifolds in
*S*^{6}satisfying Chen’s equality. J. Geom. Phys.**56**, 2279–2288 (2006)MATHMathSciNetCrossRefGoogle Scholar - 6.Harvey R., Lawson H.B.: Calibrated geometries. Acta Math.
**148**, 47–157 (1982)MATHMathSciNetCrossRefGoogle Scholar - 7.Hashimoto H., Mashimo K.: On some 3-dimensional CR submanifolds in
*S*^{6}. Nagoya Math. J.**156**, 171–185 (1999)MATHMathSciNetGoogle Scholar - 8.Sekigawa K.: Some CR submanifolds in a 6-dimensional sphere. Tensor N. S.
**41**, 13–20 (1984)MATHMathSciNetGoogle Scholar - 9.Wood R.M.W.: Framing the exceptional Lie group
*G*_{2}. Topology**15**, 303–320 (1976)MATHMathSciNetCrossRefGoogle Scholar