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.