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Ruled Real Hypersurfaces in the Complex Quadric

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Abstract

First we introduce the notions of \(\eta \)-parallel and \(\eta \)-commuting shape operator for real hypersurfaces in the complex quadric \(Q^m = SO_{m+2}/SO_mSO_2\). Next we give a complete classification of real hypersurfaces in the complex quadric \(Q^m\) with such kind of shape operators. By virtue of this classification we give a new characterization of ruled real hypersurface foliated by complex totally geodesic hyperplanes \(Q^{m-1}\) in \(Q^m\) whose unit normal vector field in \(Q^m\) is \(\mathfrak {A}\)-principal.

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Authors and Affiliations

  1. Department of Mathematics, Ibaraki University, Mito, Ibaraki, 310-8512, Japan

    Makoto Kimura

  2. Research Institute of Real and Complex Manifolds, Kyungpook National University, Daegu, 41566, Republic of Korea

    Hyunjin Lee

  3. Department of Geometry and Topology, University of Granada, Granada, 18071, Spain

    Juan de Dios Pérez

  4. Department of Mathematics and Research Institute of Real & Complex Manifolds, College of Natural Sciences, Kyungpook National University, Daegu, 41566, Republic of Korea

    Young Jin Suh

Authors
  1. Makoto Kimura
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  2. Hyunjin Lee
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  3. Juan de Dios Pérez
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  4. Young Jin Suh
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Correspondence to Young Jin Suh.

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Makoto Kimura was supported by JSPS KAKENHI Grant Number JP20K03575, Hyunjin Lee by NRF-2019-R1I1A1A-01050300, Juan de Dios Pérez by MCT-FEDER project MTM-2016-78807-C2-1-P, and Young Jin Suh by grant Proj. No. NRF-2018-R1D1A1B-05040381 from the National Research Foundation of Korea.

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Kimura, M., Lee, H., Pérez, J.d.D. et al. Ruled Real Hypersurfaces in the Complex Quadric. J Geom Anal 31, 7989–8012 (2021). https://doi.org/10.1007/s12220-020-00564-2

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  • DOI: https://doi.org/10.1007/s12220-020-00564-2

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