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Cyclic parallel structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians

Part of: Global differential geometry

Published online by Cambridge University Press:  12 August 2021

Hyunjin Lee ,
Young Jin Suh  and
Changhwa Woo
Hyunjin Lee
Affiliation:
The Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University, Daegu 41566, Repulic of Korea (lhjibis@hanmail.net)
Young Jin Suh
Affiliation:
Department of Mathematics & RIRCM, Kyungpook National University, Daegu 41566, Repulic of Korea (yjsuh@knu.ac.kr)
Changhwa Woo
Affiliation:
Department of Applied Mathematics, Pukyong National University, Busan 48513, Republic of Korea (legalgwch@pknu.ac.kr)
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Abstract

In this paper, from the property of Killing for structure Jacobi tensor $\mathbb {R}_{\xi }$, we introduce a new notion of cyclic parallelism of structure Jacobi operator $R_{\xi }$ on real hypersurfaces in the complex two-plane Grassmannians. By virtue of geodesic curves, we can give the equivalent relation between cyclic parallelism of $R_{\xi }$ and Killing property of $\mathbb {R}_{\xi }$. Then, we classify all Hopf real hypersurfaces with cyclic parallel structure Jacobi operator in complex two-plane Grassmannians.

Keywords

Hopf real hypersurfacecomplex two-plane Grassmannians(quadratic) Killing tensorcyclic parallelismstructure Jacobi operator

MSC classification

Primary: 53C40: Global submanifolds
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.)

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 939 - 964
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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