In this paper we characterize all totally $\eta$-umbilic real hypersurfaces $M$'s in complex projective or complex hyperbolic spaces by using an inequality related to the shape operator $A$ of $M$.
"Characterization of totally $\eta $-umbilic real hypersurfaces in nonflat complex space forms by some inequality." Proc. Japan Acad. Ser. A Math. Sci. 80 (5) 61 - 64, May 2004. https://doi.org/10.3792/pjaa.80.61