Bulletin of the
Korean Mathematical Society BKMS

In the paper [12] we have introduced the new kind of {\it pseudo-Einstein} ruled real hypersurfaces in complex space forms $M_n(c)$, $c{\not =}0$, which are foliated by {\it pseudo-Einstein} leaves. The purpose of this paper is to give a geometric condition for non-proper {\it pseudo-Einstein} ruled real hypersurfaces to be {\it totally geodesic} in the sense of Kimura [8] for $c>0$ and Ahn, Lee and the present author [1] for $c<0$.

Keywords: Einstein, pseudo-Einstein ruled real hypersurface, complex space form, Ricci tensor, totally geodesic

MSC numbers: Primary 53C40; Secondary 53C15