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

Abstract

It is well known that the sphere S6(1) admits an almost complex structure J, constructed using the Cayley algebra, which is nearly Kahler. 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 Kahler sphere S6(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.