Mixed foliate CR-submanifolds in a complex hyperbolic space are non-proper
A submanifold \(M\) of a Kähler manifold \(\widetilde M\) is called a CR-submanifold if (1) the maximal complex subspace \(\mathcal D_x\) of the tangent space \(T_x\widetilde M\) contained in \(T_xM\), \(x\in M\), defines a differentiable distribution \(\mathcal D\), called the holomorphic distribution, and (2) the orthogonal complementary distribution \(\mathcal D^\perp\) of \(\mathcal D\) in \(TM\) is a totally real distribution, i.e. \(J\mathcal D_x^\perp\subset T_x^\perp M,\) where \(J\) is the almost complex structure on \(\tilde M\) and \(T_x^{\perp}M\) the normal space at \(x\). Complex submanifolds and totally real submanifolds of \(\tilde M\) are trivial examples of CR-submanifolds. A CR-submanifold is said to be proper if it is neither a complex submanifold nor a totally real submanifold. It is known [see the first author and the reviewer, Isr. J. Math. 34, 353--363 (1979; Zbl 0453.53018)] that the totally real distribution \(\mathcal D^\perp\) of a CR-submanifold of a Kähler manifold is always integrable. A CR-submanifold is said to be mixed foliate if (a) the holomorphic distribution \(\mathcal D\) is integrable and (b) the second fundamental form \(\sigma^0\) of \(M\) in \(\widetilde M\) satisfies \(\sigma^0(\mathcal D,\mathcal D^\perp)=\{0\}\). The purpose of the present paper is to prove that there are no proper mixed foliate CR-submanifolds in complex hyperbolic space. The corresponding result for complex projective space was proved by \textit{A. Bejancu}, \textit{M. Kon} and \textit{K. Yano} [J. Differ. Geom. 16, 137--145 (1981; Zbl 0469.53048)]. In [ibid. 16, 305--322 (1981; Zbl 0431.53048)] the first author proved that mixed foliate CR-submanifolds of \(\mathbb{C}^m\) are CR-products and in [ibid. 16, 493--509 (1981; Zbl 0485.53051)] conjectured the present theorem.
scientific article; zbMATH DE number 4111506