Abstract
We study the hypersurfaces of \({\mathbb {S}}^3 \times {\mathbb {S}}^3\) with a specially chosen \({\mathcal {P}}\)-principal normal vector field \(\xi \), such that \(P\xi \) is collinear with \(\xi \) or \(J\xi \), where J and P are respectively the almost complex and almost product structures on \({\mathbb {S}}^3 \times {\mathbb {S}}^3\). We prove that it is impossible to have \(P\xi =\xi \) or \(P\xi =\pm J\xi \). In the case \(P\xi =-\xi \), we prove that the hypersurface is Hopf and that it has either 3 or 5 different principal curvatures. The main result is the classification of those hypersurfaces with three different principal curvatures which satisfy \(P\xi =-\xi \). Also, there is an example of family of hypersurfaces of \({\mathbb {S}}^3 \times {\mathbb {S}}^3\) with five different principal curvatures which satisfy the condition \(P\xi =-\xi \).
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The author is partially supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, project 174012.
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Djorić, M. Hypersurfaces of the Homogeneous Nearly Kähler \({\mathbb {S}}^3\times {\mathbb {S}}^3\) Whose Normal Vector Field is \({\mathcal {P}}\)-Principal. Mediterr. J. Math. 18, 251 (2021). https://doi.org/10.1007/s00009-021-01916-0
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DOI: https://doi.org/10.1007/s00009-021-01916-0
Keywords
- Nearly Kähler \({\mathbb {S}}^3\times {\mathbb {S}}^3\)
- real hypersurface
- hopf hypersurface
- \({\mathcal {P}}\)-Principal normal