\(CR\)-hypersurfaces of the six-dimensional sphere (Q1321874)

Let \(x: M\to\overline M\) be the immersion of a Riemannian submanifold \(M\) in a Riemannian manifold \(\overline M\) and let \(\nabla\) (resp. \(\overline\nabla\)) be the Riemannian connection on \(M\) (resp. \(\overline M\)). Then the Gauss formula associated with \(x\) is expressed by \(\overline\nabla_ X Y=\nabla_ X Y+h(X,Y)\); \(X,Y\in T_ pM\) where \(h\) means the second: fundamental form of \(M\) in \(\overline M\). If \(\nabla_ X h=0\), for all \(X\in T_ p(M)\), then \(h\) is said to be parallel. Assume that \((\overline M,I,g)\) is an almost Hermitian manifold. Then \(M\) is called a \(CR\)-submanifold of \(\overline M\) [\textit{A. Bejancu}, Proc. Am. Math. Soc. 69, 135-142 (1978; Zbl 0368.53040)] if there exists a holomorphic distribution \(D\) on \(M\), i.e. \(ID=D\) such that its orthogonal complement \(D^ \perp\) is totally real i.e. \(ID^ \perp\subset T^ \perp M\) (\(T^ \perp M\): normal bundle over \(M\) in \(\overline M\)). A \(CR\)-submanifold is called proper if neither \(D=0\), nor \(D^ \perp=0\). In the present paper the author proves the following theorems: Theorem 1. There does not exist a proper \(CR\)-hypersurface of the Euclidean sphere \(S^ 6\) with parallel second fundamental form. Theorem 2. \(S^ 6\) does not admit a proper \(CR\)-totally umbilical hypersurface. Theorem 3. Let \(M\) be an Einstein proper \(CR\)-hypersurface of \(S^ 6\), then \(I1\) is an extrinsic sphere.