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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}-umbilical. As a consequence, we characterize some of our new examples as those whose Reeb vector field ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi $$\end{document} is Killing.