Abstract
In this paper, we study real hypersurfaces in the complex quadric space \(Q^m\) whose structure Jacobi operator commutes with their structure tensor field. When the normal vector field is \(\mathfrak {A}\)-principal we show that the Reeb curvature \(\alpha \) is non-vanishing and determine principal curvatures of the hypersurface. In the case of \(\mathfrak {A}\)-isotropic normal vector field, we prove that the hypersurface is Hopf if it has vanishing Reeb curvature or commuting shape operator. We also consider Reeb flat hypersurfaces, namely when the Reeb curvature is zero. We see that this family of hypersurfaces is non-empty and among other results we prove that if the Ricci tensor of a Reeb flat Hopf hypersurfaces is Killing, then the Ricci tensor is parallel.
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Acknowledgements
This work was conducted during a postdoctoral fellow visit of the third author at Tarbiat Modares University with additional support from Iran national science foundation via Grant no. 95012382. The authors would like to thank Prof. Young Jin Suh for the comments we got from him. The authors would like to gratefully thank the anonymous referee for his/her invaluable comments.
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Communicated by Jost-Hinrich Eschenburg.
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The research is supported by Iran national Science foundation via Grant no. 95012382, for the second and third authors.
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Heidari, N., Kashani, S.M.B. & Vanaei, M.J. Real Hypersurfaces in \(Q^m\) with Commuting Structure Jacobi Operator. Bull. Iran. Math. Soc. 47, 351–370 (2021). https://doi.org/10.1007/s41980-020-00387-5
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DOI: https://doi.org/10.1007/s41980-020-00387-5
Keywords
- Complex quadric
- Structure Jacobi operator
- Reeb curvature
- Complex conjugation
- Kähler structure
- Killing Ricci tensor
- Killing shape operator