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Real hypersurfaces in a nonflat complex space form whose certain tensor is recurrent

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Abstract

Real hypersurfaces in a nonflat complex space form (namely, a complex projective space or a complex hyperbolic space) are interesting objects among submanifolds in Riemannian manifolds. It is known that a real hypersurface in a nonflat complex space form admits an almost contact metric structure \((\phi , \xi , \eta , g)\) induced from the ambient space. Hence we are interested in real hypersurfaces from the aspects of both submanifolds and almost contact metric manifolds. In this paper, we study real hypersurfaces in a nonflat complex space form from the viewpoint of a recurrence of the tensor field \(h(=(1/2)\mathcal {L}_\xi \phi )\). We note that the tensor h plays an important role in contact Riemannian geometry. We give a new classification which includes a special class of 3-dimensional ruled real hypersurfaces in a complex hyperbolic plane \(\mathbb {C}H^2(c)\).

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Acknowledgements

The author would like to thank Professor Yasuhiko Furihata for his valuable comments.

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  1. National Institute of Technology, Asahikawa College, Shunkodai2-2, Asahikawa, Hokkaido, 071-8142, Japan

    Kazuhiro Okumura

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  1. Kazuhiro Okumura
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Correspondence to Kazuhiro Okumura.

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Okumura, K. Real hypersurfaces in a nonflat complex space form whose certain tensor is recurrent. J. Geom. 113, 41 (2022). https://doi.org/10.1007/s00022-022-00657-z

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