Skip to main content
SpringerLink
Log in

Genuine Deformations

  • Chapter
  • First Online:
Submanifold Theory

Part of the book series: Universitext ((UTX))

  • 1497 Accesses

Abstract

In order to find necessary conditions for a submanifold in a space form with codimension greater than one to admit isometric deformations, one has to take into account that any submanifold of a deformable submanifold already possesses the isometric deformations induced by the latter. Therefore, when studying the isometric deformations of a submanifold, one should look for the “genuine” ones, that is, those which are not induced by isometric deformations of an “extended” submanifold of higher dimension. Besides, it is also of interest to consider isometric deformations of a submanifold that take place in a possibly different codimension.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 89.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Canevari, S., Tojeiro, R.: Hypersurfaces of two space forms and conformally flat hypersurfaces. Ann. Mat. Pura Appl. 197, 1–20 (2018)

    Article  MathSciNet  Google Scholar 

  2. Canevari, S., Tojeiro, R.: The Ribaucour transformation for hypersurfaces of two space forms and conformally flat hypersurfaces. Bull. Braz. Math. Soc. 49, 593–613 (2018)

    Article  MathSciNet  Google Scholar 

  3. Cartan, E.: La déformation des hypersurfaces dans l’espace conforme réel a n ≥ 5 dimensions. Bull. Soc. Math. France 45, 57–121 (1917)

    Article  MathSciNet  Google Scholar 

  4. Chion, S., Tojeiro, R.: Euclidean hypersurfaces with genuine conformal deformations in codimension two. Preprint (2018)

    Google Scholar 

  5. Dajczer, M., Florit, L.: A class of austere submanifolds. Ill. J. Math. 45, 735–755 (2001)

    Article  MathSciNet  Google Scholar 

  6. Dajczer, M., Florit, L.: Compositions of isometric immersions in higher codimension. Manuscripta Math. 105, 507–517 (2001). Erratum Manuscripta Math. 110, 135 (2003)

    Article  MathSciNet  Google Scholar 

  7. Dajczer, M., Florit, L.: Genuine deformations of submanifolds. Commun. Anal. Geom. 12, 1105–1129 (2004)

    Article  MathSciNet  Google Scholar 

  8. Dajczer, M., Florit, L.: Genuine rigidity of Euclidean submanifolds in codimension two. Geom. Dedicata 106, 195–210 (2004)

    Article  MathSciNet  Google Scholar 

  9. Dajczer, M., Gromoll, D.: Isometric deformations of compact Euclidean submanifolds in codimension 2. Duke Math. J. 79, 605–618 (1995)

    Article  MathSciNet  Google Scholar 

  10. Dajczer, M., Gromoll, D.: The Weierstrass representation for complete minimal real Kaehler submanifolds of codimension two. Invent. Math. 119, 235–242 (1995)

    Article  MathSciNet  Google Scholar 

  11. Dajczer, M., Morais, P.: Parabolic submanifolds of rank two. Mat. Contemp. 34, 195–232 (2008)

    MathSciNet  MATH  Google Scholar 

  12. Dajczer, M., Morais, P.: Isometric rigidity in codimension 2. Mich. Math. J. 58, 759–770 (2009)

    Article  MathSciNet  Google Scholar 

  13. Dajczer, M., Tojeiro, R.: On compositions of isometric immersions. J. Differ. Geom. 36, 1–18 (1992)

    Article  MathSciNet  Google Scholar 

  14. Dajczer, M., Tojeiro, R.: On submanifolds of two manifolds. Math. Z. 214, 405–413 (1993)

    Article  MathSciNet  Google Scholar 

  15. Dajczer, M., Tojeiro, R.: A rigidity theorem for conformal immersions. Indiana Univ. Math. J. 46, 491–504 (1997)

    Article  MathSciNet  Google Scholar 

  16. Dajczer, M., Tojeiro, R.: Conformal deformations of submanifolds in codimension two. J. Math. Soc. Jpn. 52, 41–50 (2000)

    Article  MathSciNet  Google Scholar 

  17. Dajczer, M., Tojeiro, R.: Submanifolds with nonparallel first normal bundle revisited. Publ. Mat. 58, 179–191 (2014)

    Article  MathSciNet  Google Scholar 

  18. Dajczer, M., Vlachos, Th.: A class of minimal submanifolds in spheres. J. Math. Soc. Jpn 69, 1197–1212 (2017)

    Article  MathSciNet  Google Scholar 

  19. Dajczer, M., Vlachos, Th.: A class of complete minimal submanifolds and their associated families of genuine deformations. Commun. Anal. Geom. 26, 699–721 (2018)

    Article  MathSciNet  Google Scholar 

  20. Dajczer, M., Florit, L., Tojeiro, R.: Euclidean hypersurfaces with genuine deformations in codimension two. Manuscripta Math. 140, 621–643 (2013)

    Article  MathSciNet  Google Scholar 

  21. do Carmo, M., Dajczer, M.: Riemannian metrics induced by two immersions. Proc. Am. Math. Soc. 86, 115–119 (1982)

    Google Scholar 

  22. Florit, L., Freitas, G.: Classification of codimension two deformations of rank two Riemannian manifolds. Commun. Anal. Geom. 25, 751–797 (2017)

    Article  MathSciNet  Google Scholar 

  23. Florit, L., Guimarães, F.: Singular genuine rigidity. Comment. Math. Helv. (2019) (to appear)

    Google Scholar 

  24. Florit, L., Tojeiro, R.: Genuine deformations of submanifolds II: the conformal case. Commun. Anal. Geom. 18, 1–23 (2010)

    Article  MathSciNet  Google Scholar 

  25. Moore, J.D.: Isometric homotopy in codimension two. Trans. Am. Math. Soc. 292, 653–663 (1985)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IMPA, Rio de Janeiro, Brazil

    Marcos Dajczer

  2. Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, São Paulo, Brazil

    Ruy Tojeiro

Authors
  1. Marcos Dajczer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruy Tojeiro
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Science+Business Media, LLC, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dajczer, M., Tojeiro, R. (2019). Genuine Deformations. In: Submanifold Theory . Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-9644-5_12

Download citation

Publish with us

Policies and ethics

Search

Navigation