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Hopf Real Hypersurfaces in the Indefinite Complex Projective Space

  • Makoto Kimura
  • Miguel OrtegaEmail author
Article

Abstract

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 \(\eta \)-umbilical. As a consequence, we characterize some of our new examples as those whose Reeb vector field \(\xi \) is Killing.

Keywords

Real hypersurface indefinite complex projective space Hopf real hypersurface 

Mathematics Subject Classification

Primary 53B25 53C50 Secondary 53C42 53B30 

Notes

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics Faculty of ScienceIbaraki UniversityMitoJapan
  2. 2.Departamento de Geometría y Topología Facultad de Ciencias Instituto de Matemáticas IEMathUGRUniversidad de GranadaGranadaSpain

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