On ruled real hypersurfaces in a complex space form (Q1327636)

A real hypersurface \(M\) of a complex space form \(N\) is said to be ruled if \(M\) is foliated by one-codimensional totally geodesic complex submanifolds of \(N\). The authors provide a sufficient condition for a real hypersurface in a non-flat complex space form to be ruled. Explicitly, let \(M\) be a connected real hypersurface in a non-flat complex space form of complex dimension \(\geq 3\) and \((\phi,\xi,\eta,g)\) the induced almost contact metric structure on \(M\). Denote by \(T_ 0\) the subbundle of \(TM\) consisting of all vectors perpendicular to \(\xi\) and by \(A\) the shape operator of \(M\). If \(g((\nabla_ X A)Y,Z)= 0\) and \(g((A\phi- \phi A)X,Y)= 0\) for all \(X,Y,Z\in T_ 0\) and if \(\xi\) is not a principal curvature vector of \(M\) somewhere, then \(M\) is ruled. The authors also discuss a particular example of a minimal ruled real hypersurface in complex hyperbolic space.