Real hypersurfaces of a complex hyperbolic space satisfying \(L_\xi S=0\) (Q2755652)

The authors' improve a result by \textit{U-Hang Ki}; \textit{Nam Gil Kim} and \textit{Sung-Baik Lee} [J. Korean Math. Soc. 29, 63-77 (1992; Zbl 0753.53035)]\ about real hypersurfaces of a complex hyperbolic space \(\mathbb CH_\mu\) which satisfy \(L_\xi S=0\), where \(S\) is the Ricci tensor and \(L_\xi\) is the Lie derivative with respect to the structure vector \(\xi\). The following theorem is proved:NEWLINENEWLINENEWLINELet \(M\) be a connected real hypersurface of \(\mathbb CH_\mu\). If \(M\) satisfies \(L_\xi S=0\) and \(A_\xi\) (where \(A\) is the shape operator) is a principal curvature vector on \(M\), then \(\xi\) is a principal curvature vector on \(M\).