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 Kähler. Let M be a Riemannian submanifold of a manifold (Formula Presented.) 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 D1 in the tangent bundle such that its orthogonal complement D2 in the tangent bundle is totally real. If the second fundamental form vanishes on Di, the submanifold is Di-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 D1- and D2-geodesic. Here, we investigate the class of the three-dimensional minimal CR submanifolds M of the nearly Kähler sphere S6(1) which are not linearly full. We show that this class coincides with the class of D1- and D2- 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 D1- and D2-geodesic CR submanifolds.