Characterizations of ruled real hypersurfaces in 2-dimensional complex space form (Q2796850)

A real hypersurface \(M\) in a complex space form \(M_n(c),\) \(c\neq 0\) has an almost contact metric structure \((\phi,g,\xi,\eta)\) induced by the Kähler metric.NEWLINENEWLINEIn this paper, the authors give one geometrical characterization of ruled real hypersurface \(M\) in \(2\)-dimensional complex space form \(M_2(c),\) \(c\neq 0\) in terms of the \(\eta\)-parallel second fundamental form and the integrable holomorphic distribution \(\mathbb{D}.\) This distribution is defined by \(\mathbb{D}(p)=\{X \in T_pM \mid g(X, \xi)=0\},\) where \(T_pM \) in the tangent space of \(M\) and \(\xi\) is the Reeb vector field.NEWLINENEWLINEThe main result here is: ``Let \(M\) be a real hypersurface in a complex space form \(M_2(c),\) \(c\neq 0.\) Then the second fundamental form of \(M\) is \(\eta\)-parallel and the holomorphic distribution \(\mathbb{D}\) is integrable if and only if \(M\) is locally congruent to a ruled real hypersurface.''