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\(\mathfrak D \)-parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians. (English) Zbl 1292.53037

Summary: In this paper, we give non-existence theorems for Hopf hypersurfaces in complex two-plane Grassmannians \(G_2(\mathbb{C }^{m+2})\) with \(\mathfrak D \)-parallel normal Jacobi operator \({\bar{R}}_N\) and \(\mathfrak D \)-parallel structure Jacobi operator \(R_{\xi }\) if the \(\mathfrak D \) or \(\mathfrak D ^{\bot}\)-component of the Reeb vector field is invariant by the shape operator, respectively.

MSC:

53C40 Global submanifolds
53C30 Differential geometry of homogeneous manifolds
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