Classifications of real hypersurfaces in complex space forms by means of curvature conditions (Q1420713)

Let \(M\) be a connected real hypersurface with unit normal vector field \(N\) in a complex space form \(M^n(c)\) of complex dimension \(n\geq 3\), of constant curvature \(c\neq 0\), and with almost complex structure \(J\). For such an \(M\) the following two conditions are considered: \[ \begin{gathered} Q(X,Y)Z+ Q(Z,X)Y+ Q(Y, Z)X= 0,\tag{1}\\ g(Q(X,Y) Z,W)+ g(Q(Y, Z)X,W)+ g(Q(Z, X)Y, W)= 0,\tag{2}\end{gathered} \] for any \(X\), \(Y\), \(Z\), \(W\) orthogonal to \(\xi= -JN\), where \(Q(X,Y)Z= (R(X, Y)A)Z\) and \(A\) is the Weingarten endomorphism. It is proved that the following statements are pairwise equivalent: (i) \(M\) satisfies (1); (ii) \(M\) satisfies (2); (iii) \(M\) is either a ruled real hypersurface or an open subset of a real hypersurface of \(A_0\) or \(A_1\) (i.e. a horosphere of \(\text{CH}^n\) or a tube over a totally geodesic submanifold). Meanwhile the author shows that a result by \textit{Y. Matsuyama} [Yokohama Math. J. 46, No.~2, 119--126 (1999; Zbl 0980.53028)] is not correct. Also some characterizations for real hypersurfaces of type \(A_0\) and \(A_1\) are given. A corollary is made that there are no semiparallel (i.e. with \(R\cdot A= 0\)) hypersurfaces in \(\overline M^n(c)\), \(n\geq 2\), \(c\neq 0\). This solves the problem, which remained open in the paper by \textit{R. Niebergall} and \textit{P. J. Ryan} [Kyungpook Math. J. 38, No.~1, 227--234 (1998; Zbl 0906.53015)].