Mediterranean Journal of Mathematics

, Volume 12, Issue 4, pp 1429–1449 | Cite as

Three-Dimensional Minimal CR Submanifolds of the Sphere S 6 (1) Contained in a Hyperplane

  • Miroslava Antić
  • Luc Vrancken


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.


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

Mathematics Subject Classification

Primary 53B20 Secondary 53B21 53B25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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. 2.
    Bejancu, A.: Geometry of CR-Submanifolds. D. Reidel Publ., Dordrecht (1986)Google Scholar
  3. 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. 4.
    Chen B.Y.: Some pinching and classification theorems for minimal submanifolds. Archiv. Math. (Basel) 60, 568–578 (1993)MATHCrossRefGoogle Scholar
  5. 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. 6.
    Harvey R., Lawson H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Hashimoto H., Mashimo K.: On some 3-dimensional CR submanifolds in S 6. Nagoya Math. J. 156, 171–185 (1999)MATHMathSciNetGoogle Scholar
  8. 8.
    Sekigawa K.: Some CR submanifolds in a 6-dimensional sphere. Tensor N. S. 41, 13–20 (1984)MATHMathSciNetGoogle Scholar
  9. 9.
    Wood R.M.W.: Framing the exceptional Lie group G 2. Topology 15, 303–320 (1976)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia
  2. 2.LAMAVUniversité de ValenciennesValenciennes Cedex 9France
  3. 3.Departement WiskundeKU LeuvenLeuvenBelgium

Personalised recommendations