A certain \(\eta\)-parallelism on real hypersurfaces in a nonflat complex space form. (English) Zbl 1515.53018
Summary: In this paper, we give the complete classification of real hypersurfaces in a nonflat complex space form from the viewpoint of the \(\eta \)-parallelism of the tensor field \(h(=(1/2)\mathcal{L}_\xi \varphi)\). In addition we investigate real hypersurfaces whose tensor \(h\) is either Killing type or transversally Killing tensor. In particular, we shall determine Hopf hypersurfaces whose tensor \(h\) is transversally Killing tensor by using an application of the classification of real hypersurfaces admitting \(\eta\)-parallelism with respect to the tensor \(h\).
MSC:
| 53B25 | Local submanifolds |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 53D15 | Almost contact and almost symplectic manifolds |
| 32V40 | Real submanifolds in complex manifolds |
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