Geodesic
A characterization of totally $\eta$-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1279-1287.

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We give a characterization of totally $\eta $-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator $A$ of a real hypersurface $M$ of a complex space form $M^n(c)$, $c\neq 0$, $n\geq 3$, satisfies $g(AX,Y)=ag(X,Y)$ for any $X,Y\in T_0(x)$, $a$ being a function, where $T_0$ is the holomorphic distribution on $M$, then $M$ is a totally $\eta $-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of $\eta $-umbilical real hypersurfaces.
Classification : 53C25, 53C40, 53C55
Keywords: real hypersurface; totally $\eta $-umbilical real hypersurface; ruled real hypersurface 
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     title = {A characterization of totally $\eta$-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form},
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Kon, Mayuko. A characterization of totally $\eta$-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 4, pp. 1279-1287. http://geodesic.mathdoc.fr/item/CMJ_2008__58_4_a30/