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Commuting Jacobi Operators on Real Hypersurfaces of Type B in the Complex Quadric

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Abstract

In this paper, first, we investigate the commuting property between the normal Jacobi operator \({\bar R}_{N}\) and the structure Jacobi operator Rξ for Hopf real hypersurfaces in the complex quadric Qm = SOm+ 2/SOmSO2 for \(m \geqslant 3\), which is defined by \({\bar R}_{N} R_{\xi } = R_{\xi }{\bar R}_{N}\). Moreover, a new characterization of Hopf real hypersurfaces with \(\mathfrak A\)-principal singular normal vector field in the complex quadric Qm is obtained. By virtue of this result, we can give a remarkable classification of Hopf real hypersurfaces in the complex quadric Qm with commuting Jacobi operators.

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Acknowledgments

The authors would like to express their sincere gratitude to the reviewers and the editors for their valuable comments throughout the manuscript. Their efforts have allowed us to greatly improve the manuscript.

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Authors and Affiliations

  1. The Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University, Daegu, 41566, Korea

    Hyunjin Lee

  2. Department of Mathematics & RIRCM, Kyungpook National University, Daegu, 41566, Korea

    Young Jin Suh

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  1. Hyunjin Lee
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  2. Young Jin Suh
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Correspondence to Hyunjin Lee.

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The first author was supported by grant Proj. No. NRF-2019-R1I1A1A01050300 and the second author by NRF-2018-R1D1A1B05040381 from National Research Foundation of Korea.

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Lee, H., Suh, Y.J. Commuting Jacobi Operators on Real Hypersurfaces of Type B in the Complex Quadric. Math Phys Anal Geom 23, 44 (2020). https://doi.org/10.1007/s11040-020-09370-2

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