On $C^2$ umbilical hypersurfaces
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- by Carlo Mantegazza
- Proc. Amer. Math. Soc. 150 (2022), 1745-1748
- DOI: https://doi.org/10.1090/proc/15804
- Published electronically: January 5, 2022
Abstract:
We show by an elementary argument that the second fundamental form of a connected, totally umbilical hypersurface of class $C^2$ is a constant multiple of the metric tensor. It follows that the hypersurface is smooth and it is either a piece of a hyperplane or of a sphere.References
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- Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1969 original; A Wiley-Interscience Publication. MR 1393941
- Michael Spivak, A comprehensive introduction to differential geometry. Vol. III, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532832
Bibliographic Information
- Carlo Mantegazza
- Affiliation: Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Cintia, Monte S. Angelo 80126 Napoli, Italy
- MR Author ID: 611900
- ORCID: 0000-0002-0658-0339
- Email: c.mantegazza@sns.it
- Received by editor(s): July 13, 2021
- Published electronically: January 5, 2022
- Communicated by: Jiaping Wang
- © Copyright 2022 by Carlo Mantegazza
- Journal: Proc. Amer. Math. Soc. 150 (2022), 1745-1748
- MSC (2020): Primary 53A07, 53C42
- DOI: https://doi.org/10.1090/proc/15804
- MathSciNet review: 4375761