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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The infinitesimal deformations of hypersurfaces that preserve the Gauss map
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by Marcos Dajczer and Miguel Ibieta Jimenez;
Proc. Amer. Math. Soc. 152 (2024), 3565-3573
DOI: https://doi.org/10.1090/proc/16784
Published electronically: June 5, 2024

Abstract:

Classifying the nonflat hypersurfaces in Euclidean space $f\colon M^n\to \mathbb {R}^{n+1}$ that locally admit smooth infinitesimal deformations that preserve the Gauss map infinitesimally was a problem only considered by Schouten in 1928 [Proceedings Amsterdam 31 (1928), pp. 208–218]. He found two conditions that are necessary and sufficient, with the first one being the minimality of the submanifold. The second is a technical condition that does not clarify much about the geometric nature of the hypersurface. In that respect, the parametric solution of the problem given in this note yields that the submanifold has to be Kaehler.
References
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Bibliographic Information
  • Marcos Dajczer
  • Affiliation: Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, E-30100 Espinardo, Murcia, Spain
  • MR Author ID: 54140
  • ORCID: 0000-0003-2832-6849
  • Email: marcos@impa.br
  • Miguel Ibieta Jimenez
  • Affiliation: Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP 13566-590, Brazil
  • ORCID: 0000-0001-7967-1058
  • Email: mibieta@icmc.usp.br
  • Received by editor(s): September 30, 2023
  • Received by editor(s) in revised form: December 20, 2023, and January 1, 2024
  • Published electronically: June 5, 2024
  • Additional Notes: The first author was partially supported by the grant PID2021-124157NB-I00 funded by MCIN/AEI/10.13039/501100011033/ ‘ERDF A way of making Europe’, Spain, and was also supported by Comunidad Autónoma de la Región de Murcia, Spain, within the framework of the Regional Programme in Promotion of the Scientific and Technical Research (Action Plan 2022), by Fundación Séneca, Regional Agency of Science and Technology, REF, 21899/PI/22. The seconda author was supported by FAPESP with the grant 2022/05321-9.
  • Communicated by: Jiaping Wang
  • © Copyright 2024 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 3565-3573
  • MSC (2020): Primary 53A07, 53B25
  • DOI: https://doi.org/10.1090/proc/16784
  • MathSciNet review: 4767284