
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.
real hypersurface, Local submanifolds, almost contact metric structure, minimal ruled real hypersurface, Global differential geometry of Hermitian and Kählerian manifolds, complex hyperbolic space
real hypersurface, Local submanifolds, almost contact metric structure, minimal ruled real hypersurface, Global differential geometry of Hermitian and Kählerian manifolds, complex hyperbolic space
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