Annali di Matematica Pura ed Applicata (1923 -)

, Volume 196, Issue 6, pp 1961–1979 | Cite as

The infinitesimally bendable Euclidean hypersurfaces

Article

Abstract

The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space.

Keywords

Infinitesimal bending Euclidean hypersurface Envelope of hyperplanes 

Mathematics Subject Classification

53B25 53A07 

References

  1. 1.
    Beez, R.: Zur Theorie des Krümmungsmasses von Mannigfaltigkeiten höhere Ordnung. Zeit. für Math. und Physik 21, 373–401 (1876)MATHGoogle Scholar
  2. 2.
    Bianchi, L.: Sulle varietà a tre dimensioni deformabili entro lo spazio euclideo a quattro dimensioni. Memorie di Matematica e di Fisica della Società Italiana delle Scienze, serie III 13, 261–323 (1905)MATHGoogle Scholar
  3. 3.
    Cartan, E.: La déformation des hypersurfaces dans l’espace euclidien réel a \(n\) dimensions. Bull. Soc. Math. Fr. 44, 65–99 (1916)CrossRefMATHGoogle Scholar
  4. 4.
    Cesaro, E.: Lezioni di Geometria intrinseca. Tipografia della R. Accademia delle scienze, Napoli (1896)Google Scholar
  5. 5.
    Dajczer, M., Florit, L., Tojeiro, R.: On deformable hypersurfaces in space forms. Ann. Mat. Pura Appl. 174, 361–390 (1998)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Dajczer, M., Florit, L., Tojeiro, R.: Euclidean hypersurfaces with genuine deformations in codimension two. Manuscripta Math. 140, 621–643 (2013)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Dajczer, M., Gromoll, D.: Gauss parametrizations and rigidity aspects of submanifolds. J. Differ. Geom. 22, 1–12 (1985)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Dajczer, M., Rodríguez, L.: Infinitesimal rigidity of Euclidean submanifolds. Ann. Inst. Fourier 40, 939–949 (1990)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Killing, W.: Die nicht-Euklidischen Raumformen in Analytische Behandlung. Teubner, Leipzig (1885)MATHGoogle Scholar
  10. 10.
    Sbrana, U.: Sulla deformazione infinitesima delle ipersuperficie. Ann. Mat. Pura Appl. 15, 329–348 (1908)CrossRefMATHGoogle Scholar
  11. 11.
    Sbrana, U.: Sulla varietá ad \(n-1\) dimensioni deformabili nello spazio euclideo ad \(n\) dimensioni. Rend. Circ. Mat. Palermo 27, 1–45 (1909)CrossRefMATHGoogle Scholar
  12. 12.
    Schouten, J.A.: On infinitesimal deformations of \(V^m\) in \(V^n\). In: Proceedings Amsterdam, vol. 36, pp. 1121–1131 (1928)Google Scholar
  13. 13.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry. Publish or Perish Inc., Houston (1979)MATHGoogle Scholar
  14. 14.
    Struik, D.J.: Grundzüge der Mehrdimensionalen Differentialgeometrie: in Direkter Darstellung. Springer, Berlin (1922)CrossRefMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil
  2. 2.Mathematics DepartmentUniversity of IoanninaIoanninaGreece

Personalised recommendations