Abstract
Urbano introduced C function for hypersurfaces in \({\mathbb {S}}^{2} \times {\mathbb {S}}^{2}\) and used it to classify isoparametric hypersurfaces. In this paper, we first show that for a hypersurface in \({\mathbb {S}}^{2} \times {\mathbb {S}}^{2}\) with \(|C|<1\), the first fundamental form and Urbano’s C function determines its second fundamental form. Moreover, we derive examples of hypersurfaces with \(C\equiv c_0\) for any \(|c_0|<1\). Note that all examples with constant C shown in Urbano’s paper have \(C\equiv 0\) or \(\pm 1\).
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Peng Wang was supported by NSFC Project 11971107. Xiaozhen Wang was supported by NSFC Project 11971107 and the Natural Science Foundation of Fujian Province (No. 2022J02028).
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Lu, X., Wang, P. & Wang, X. On the Geometry of Hypersurfaces in \({\mathbb {S}}^{2} \times {\mathbb {S}}^{2}\). Results Math 78, 10 (2023). https://doi.org/10.1007/s00025-022-01787-1
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DOI: https://doi.org/10.1007/s00025-022-01787-1