Real hypersurfaces of a complex projective space in terms of holomorphic distribution (Q2640898)

The authors study real hypersurfaces in complex projective space \({\mathbb{C}}P^ m\) from the point of view of holomorphic distributions. For a real hypersurface M of \({\mathbb{C}}P^ m\) let \(T^ 0M\) denote the maximal subbundle of the tangent bundle TM of M which is invariant with respect to the complex structure of \({\mathbb{C}}P^ m\). \(T^ 0M\) is a holomorphic subbundle of TM and of real codimension one. Let \(\nabla^ 0\) be the induced connection on \(T^ 0M\) and \(A^ 0\) be the restriction to \(T^ 0M\) of the second fundamental form of \(T^ 0M\) regarded as a subbundle in \(T{\mathbb{C}}P^ m\) (along M). \(A^ 0\) is said to be \(\eta\)-parallel if \(\nabla^ 0_ XA^ 0=0\) for all sections X in \(T^ 0M.\) The main purpose of the paper is to provide that any real hypersurface in \({\mathbb{C}}P^ m\) for which \(A^ 0\) is \(\eta\)-parallel is locally congruent to one of the following: (1) a geodesic hypersphere in \({\mathbb{C}}P^ m\), (2) a tube about a totally geodesic \({\mathbb{C}}P^ k\) in \({\mathbb{C}}P^ m\) \((0<k<m-2)\), (3) a tube about the standard embedding of the complex quadric in \({\mathbb{C}}P^ m\), (4) a ruled real hypersurface in \({\mathbb{C}}P^ m\), that is, a real hypersurface in \({\mathbb{C}}P^ m\) for which \(T^ 0M\) is integrable and any integral manifold is a totally geodesic \({\mathbb{C}}P^{m-1}\), (5) a real hypersurface in \({\mathbb{C}}P^ m\) for which \(T^ 0M\) is integrable and any integral manifold is a complex quadric.