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# Lie derivatives and Ricci tensor on real hypersurfaces in complex two-plane Grassmannians

• Imsoon Jeong,
Division of Future Capability Education, Ju Si-Gyeong College, Pai Chai University, Republic of Korea
• Juan de Dios Pérez,
• Young Jin Suh,
Department of Mathematics and Research Institute of Real and Complex Manifold, Kyungpook National University, Republic of Korea
• Changhwa Woo,
Department of Mathematics Education, Woosuk University, Republic Of Korea
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## Abstract

On a real hypersurface $M$ in a complex two-plane Grassmannian $G_2({\mathbb C}^{m+2})$ we have the Lie derivation ${\mathcal L}$ and a differential operator of order one associated to the generalized Tanaka-Webster connection $\widehat {\mathcal L} ^{(k)}$. We give a classification of real hypersurfaces $M$ on $G_2({\mathbb C}^{m+2})$ satisfying $\widehat {\mathcal L} ^{(k)}_{\xi}S={\mathcal L}_{\xi}S$, where $\xi$ is the Reeb vector field on $M$ and $S$ the Ricci tensor of $M$.
 Keywords: real hypersurface, complex two-plane Grassmannian, Hopf hypersurface, shape operator, Ricci tensor, Lie derivation
 MSC Classifications: 53C40 - Global submanifolds [See also 53B25] 53C15 - General geometric structures on manifolds (almost complex, almost product structures, etc.)

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