On real hypersurfaces with \(\eta\)-parallel curvature tensor in complex space forms
Let \(M\) be a real hypersurface of a complex space form \(M_n(c)\), \(n\geq 3\), and \((\phi,\xi,\eta,g)\) the induced almost contact metric structure of \(M\). The condition \(g((\nabla_XR)(Y,Z)U,V)=0\) for any \(X\), \(Y\), \(Z\), \(U\), and \(V\) in the distribution \(T_0\) orthogonal to \(\xi\) characterizes the \(\eta\)-parallelity of the curvature tensor \(R\) of \(M\). It is proved that such \(M\), which also satisfies the condition \(g((A\phi-\phi A)X,Y)=0\) for the second fundamental tensor \(A\), is locally congruent to one of the following: (1) for \(c>0\) \((A_1)\) a tube over a hyperplane, \((A_2)\) a tube over a totally geodesic \(P_k(\mathbb{C})\) \((1\leq k\leq n-2)\), \((R)\) a ruled real hypersurface foliated by a totally geodesic complex hypersurface \(P_{n-1}(\mathbb{C})\); (2) for \(c<0\) \((A_0)\) a horosphere in \(H_n(\mathbb{C})\), \((A_1)\) a tube of a totally geodesic hyperplane \(H_k(\mathbb{C})\) (\(k=0\) or \(n-1\)), \((A_2)\) a tube of a totally geodesic \(H_k(\mathbb{C})\) \((1\leq k\leq n-2)\), \((R)\) a ruled real hypersurface foliated by totally geodesic complex hypersurfaces \(H_{n-1}(\mathbb{C})\). This generalizes the results by \textit{M. Okumura} [Trans. Am. Math. Soc. 212, 355--364 (1975; Zbl 0288.53043)] and by \textit{S. Montel} and \textit{A. Romero} [Geom. Dedicata 20, 245--261 (1986; Zbl 0587.53052)].
scientific article; zbMATH DE number 2099080