The infinitesimal deformations of hypersurfaces that preserve the Gauss map
HTML articles powered by AMS MathViewer
- 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
- HTML | PDF | Request permission
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
- L. Bianchi, Sulle varietà a tre dimensioni deformabili entro lo spazio euclideo a quattro dimensioni, Mem. Soc. It. delle Sc. III, XIII (1905), 261–323.
- Eugenio Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1–23. MR 57000, DOI 10.2307/1969817
- E. Cartan, La déformation des hypersurfaces dans l’espace euclidien réel à $n$ dimensions, Bull. Soc. Math. France 44 (1916), 65–99 (French). MR 1504750, DOI 10.24033/bsmf.964
- E. Cartan, La déformation des hypersurfaces dans l’espace conforme réel à $n \ge 5$ dimensions, Bull. Soc. Math. France 45 (1917), 57–121 (French). MR 1504762, DOI 10.24033/bsmf.975
- E. Cesàro, Lezioni di Geometria intrinseca, Napoli, 1896.
- M. Dajczer, L. Florit, and R. Tojeiro, On deformable hypersurfaces in space forms, Ann. Mat. Pura Appl. (4) 174 (1998), 361–390. MR 1746935, DOI 10.1007/BF01759378
- Marcos Dajczer and Detlef Gromoll, Real Kaehler submanifolds and uniqueness of the Gauss map, J. Differential Geom. 22 (1985), no. 1, 13–28. MR 826421
- Marcos Dajczer and Detlef Gromoll, Euclidean hypersurfaces with isometric Gauss maps, Math. Z. 191 (1986), no. 2, 201–205. MR 818664, DOI 10.1007/BF01164024
- Marcos Dajczer and Detlef Gromoll, The Weierstrass representation for complete minimal real Kaehler submanifolds of codimension two, Invent. Math. 119 (1995), no. 2, 235–242. MR 1312499, DOI 10.1007/BF01245181
- Marcos Dajczer and Miguel Ibieta Jimenez, Infinitesimal variations of submanifolds, Ensaios Matemáticos [Mathematical Surveys], vol. 35, Sociedade Brasileira de Matemática, Rio de Janeiro, [2021] ©2021. MR 4448588
- Marcos Dajczer and Miguel Ibieta Jimenez, A construction of Sbrana-Cartan hypersurfaces in the discrete class, Proc. Amer. Math. Soc. 151 (2023), no. 7, 3069–3081. MR 4579379, DOI 10.1090/proc/16390
- M. Dajczer, M. I. Jimenez, and Th. Vlachos, Conformal infinitesimal variations of Euclidean hypersurfaces, Ann. Mat. Pura Appl. (4) 201 (2022), no. 2, 743–768. MR 4386843, DOI 10.1007/s10231-021-01136-z
- Marcos Dajczer and Lucio Rodríguez, Rigidity of real Kaehler submanifolds, Duke Math. J. 53 (1986), no. 1, 211–220. MR 835806, DOI 10.1215/S0012-7094-86-05314-7
- M. Dajczer and L. Rodriguez, Infinitesimal rigidity of Euclidean submanifolds, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 4, 939–949 (1991) (English, with French summary). MR 1096598, DOI 10.5802/aif.1242
- Marcos Dajczer and Ruy Tojeiro, Submanifold theory, Universitext, Springer, New York, 2019. Beyond an introduction. MR 3969932, DOI 10.1007/978-1-4939-9644-5
- M. Dajczer and Th. Vlachos, The infinitesimally bendable Euclidean hypersurfaces, Ann. Mat. Pura Appl. (4) 196 (2017), no. 6, 1961–1979. MR 3714748, DOI 10.1007/s10231-017-0641-8
- G. Darboux, Leçons sur la théorie générale des surfaces I, II, Gauthier-Villars, Paris, 1941.
- Peter Hennes, Weierstrass representations of minimal real Kahler submanifolds, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–State University of New York at Stony Brook. MR 2703366
- U. Sbrana, Sulla deformazione infinitesima delle ipersuperficie, Ann. Mat. Pura Appl. 15 (1908), 329–348.
- U. Sbrana, Sulle varietà ad $n-1$ dimensioni deformabili nello spazio euclideo ad $n$ dimensioni, Rend. Circ. Mat. Palermo 27 (1909), 1–45.
- J.A. Schouten, On infinitesimal deformations of $V^m$ in $V^n$. Proceedings Amsterdam 31 (1928), 208–218.
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