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