Abstract
In this paper, geometric characterizations of conformally flat and radially flat hypersurfaces in \(\mathbb {S}^n \times \mathbb {R}\) and \(\mathbb {H}^n \times \mathbb {R}\) are given by means of their extrinsic geometry. Under suitable conditions on the shape operator, we classify conformally flat hypersurfaces in terms of rotation hypersurfaces. In addition, a close relation between radially flat hypersurfaces and semi-parallel hypersurfaces is established. These results lead to geometric descriptions of hypersurfaces with special intrinsic structures, such as Einstein metrics, Ricci solitons and hypersurfaces with constant scalar curvature.
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Rafael Novais was supported by CNPq.
João Paulo dos Santos was supported by FAPDF 0193.001346/2016.
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Novais, R., dos Santos, J.P. Intrinsic and extrinsic geometry of hypersurfaces in \(\varvec{\mathbb {S}}^{\varvec{n}} \varvec{\times } \varvec{\mathbb {R}}\) and \(\varvec{\mathbb {H}}^{\varvec{n}}\varvec{\times } \varvec{\mathbb {R}}\) . J. Geom. 108, 1115–1127 (2017). https://doi.org/10.1007/s00022-017-0399-6
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DOI: https://doi.org/10.1007/s00022-017-0399-6