It is well known that the sphere S 6 (1) admits an almost complex structure J which is nearly Kähler. A submanifold M of an almost Hermitian manifold is called a CR submanifold if it admits a differentiable almost complex distribution D 1 such that its orthogonal complement is a totally real distribution. In this case the normal bundle of the submanifold also splits into two distributions D 3 , which is almost complex, and a totally real complement. In the case of the proper threedimensional CR

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... submanifold of a six-dimensional manifold the distribution D 3 is non-trivial. Here, we investigate three-dimensional CR submanifolds of the sphere S 6 (1) admitting an umbilic direction orthogonal to D 3 and show that such submanifolds do not exist.