On semi-Ryan complex submanifolds in an indefinite complex space form
Recently the present authors have given a complete classification of semisymmetric complex hypersurfaces, i.e. satisfying \(R(X,Y) \cdot R=0\), in a semidefinite complex space form \(M^m_1(c)\) [see the same journal, 31, 417-435 (2001)]. Now they give a complete classification of \(n\)-dimensional complete space-like complex submanifolds \(M\), satisfying \(R(X,Y) \cdot S=0\), where \(S\) is the Ricci tensor of \(M\) (such an \(M\) is called here semi-Ryan), in an \((n+p)\)-dimensional indefinite complex space form \(M^{n+p}_{0 +t}(c)\) of index \(2t\). It is shown that if the totally real holomorphic bisectional curvature of such an \(M\) is nonvanishing at all of its points, then \(M\) is Einstein. (The case of hypersurface has been investigated by \textit{A. Romero}, Math. Z. 192, 627-635 (1986; Zbl 0579.53019). In the real case the hypersurfaces with \(R(X,Y)\cdot S=0\) are classified by \textit{V. A. Mirzoyan}, Sb. Math. 191, 1323-1338 (2000; Zbl 1001.53006), who called them Ric-semiparallel). For an \(n\)-dimensional space-like semi-Ryan complex hypersurface \(M\) of the complex Minkowski space \(C^{n+1}_1\) the authors prove here that such an \(M\) is cylindrical.
scientific article; zbMATH DE number 1744056
scientific article