Abstract
We introduce the notion of normal Jacobi operator of Codazzi type for real hypersurfaces in the complex quadric \(Q^m = \text {SO}_{m+2}/\text {SO}_m\text {SO}_2\) . The normal Jacobi operator of Codazzi type implies that the unit normal vector field N becomes \({\mathfrak {A}}\)-principal or \({\mathfrak {A}}\)-isotropic. Then, according to each case, we give a complete classification of Hopf real hypersurfaces in \(Q^m = \text {SO}_{m+2}/\text {SO}_m\text {SO}_2\) with normal Jacobi operator of Codazzi type. The result of the classification is that no such hypersurfaces exist.
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Acknowledgements
The present authors would like to express their deep gratitude to the referees for their careful reading of our paper and useful comments to develop the first version of this manuscript.
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Communicated by V. Ravichandran.
This work was supported by Grant Proj. No. NRF-2015-R1A2A1A-01002459 from National Research Foundation of Korea, and the first author by NRF-2017-R1A2B4005317.
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Jeong, I., Kim, G.J. & Suh, Y.J. Real Hypersurfaces in the Complex Quadric with Normal Jacobi Operator of Codazzi Type. Bull. Malays. Math. Sci. Soc. 41, 945–964 (2018). https://doi.org/10.1007/s40840-017-0485-9
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DOI: https://doi.org/10.1007/s40840-017-0485-9
Keywords
- Normal Jacobi operator of Codazzi type
- \({\mathfrak {A}}\)-isotropic
- \({\mathfrak {A}}\)-principal
- Kähler structure
- Complex conjugation
- Complex quadric