Hopf Real Hypersurfaces in the Indefinite Complex Projective Space

We wish to attack the problems that Anciaux and Panagiotidou posed in (Differ Geom Appl 42:1–14, https://doi.org/10.1016/j.difgeo.2015.05.004 , 2015 ), for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors’ point of view, obtaining cleaner equations for the almost-contact metric structure. To make the theory meaningful, we construct new families of non-degenerate Hopf real hypersurfaces whose shape operator is diagonalisable, and one Hopf example with degenerate metric and non-diagonalisable shape operator. Next, we obtain a rigidity result. We classify those real hypersurfaces which are η -umbilical. As a consequence, we characterize some of our new examples as those whose Reeb vector field ξ is Killing.