On semi-symmetric complex hypersurfaces of a semi-definite complex space form (Q5948860)

The authors prove the following classification theorems:NEWLINENEWLINENEWLINETheorem 1. Let \(M^n_s\) be an \(n\)-dimensional semi-symmetric and semi-definite complex hypersurface of index \(2s\) in \(M =M^{n+1}_{s+t} (c)\), \(0\leq s\leq n\), \(t= 0\) or 1, \(c\neq 0\). Then \(M\) is totally geodesic with scalar curvature \(r=n (n+1) c\) or Einstein with scalar curvature \(r=n^2c\).NEWLINENEWLINENEWLINETheorem 2. Let \(M\) be an \(n\)-dimensional semi-symmetric complex hypersurface of \(C^{n+1}_t\), \(t=0\) or 1. If it has no geodesic points then for any point \(x\) in \(M\) there exists a totally geodesic hypersurface \(M(x)\) of \(M\) through \(x\).