Skip to main content
Log in

Bianchi–Bäcklund transforms and dressing actions, revisited

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We characterize Bianchi–Bäcklund transformations of surfaces of positive constant Gauss curvature in terms of dressing actions of the simplest type on the space of harmonic maps.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bergvelt M.J., Guest M.A.: Action of loop groups on harmonic maps. Trans. Am. Math. Soc. 326, 861–886 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bianchi, L.: Lezioni di geometria differenziale, Piza (1902)

  3. Burstall, F.E.: Isothermic surfaces: conformal geometry, clifford algebra and integrable systems. In: Terng, C.-L. (ed.) Integrable Systems, Geometry and Topology, vol. 36, pp. 1–82. AMS/IP Studies in Advanced Math. (2006)

  4. Burstall, F.E., Pedit, F.: Harmonic maps via Adler-Konstant-Symes theory. In: Fordy, A.P., Wood, J.C. (eds.) Harmonic Maps and Integrable Systems, Aspects of Mathematics E23, CMP 94:09, pp. 221–272. Vieweg (1994)

  5. Burstall F.E., Pedit F.: Dressing orbits of harmonic maps. Duke Math. J. 80, 353–382 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Donaldson, N., Fox, D., Goertsches, O.: Generators for rational loop groups and geometric applications. arXiv:0803.0029v1 [math.DG]

  7. Eisenhart L.P.: A treatise on the differential geometry of curves and surfaces. Dover, New York (1960)

    MATH  Google Scholar 

  8. Hélein, F.: Constant mean curvature surfaces, harmonic maps and integrable systems, Lectures in Mathematics: ETH Zürich, Birkhäuser (2001)

  9. Hertrich-Jeromin, U., Pedit, F.: Remarks on the Darboux tranform of isothermic surfaces. Doc. Math. 2 (1997)

  10. Kobayashi S., Inoguchi J.: Characterizations of Bianchi–Bäcklund transformations of constant mean curvature surfaces. Int. J. Math. 16(2), 101–110 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Mahler, A.: Bianchi–Bäcklund and dressing transformations on constant mean curvature surfaces, Ph.D. thesis, University of Toledo (2002)

  12. Melko M., Sterling I.: Application of soliton theory to the construction of pseudospherical surfaces in \({\mathbb{R}^3}\). Ann. Glob. Anal. Geom. 11, 65–107 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Sterling I., Wente H.: Existence and classification of constant mean curvature multibubbletons of finite and infinite type. Indiana Univ. Math. J. 42(4), 1239–1266 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Terng C.-L., Uhlenbeck K.: Geometry of solitons. Not. Am. Math. Soc. 47(1), 339–403 (2000)

    MathSciNet  Google Scholar 

  15. Terng C.-L., Uhlenbeck K.: Bäcklund transformations and loop group actions. Comm. Pure Appl. Math. 53, 1–75 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Uhlenbeck K.: Harmonic maps into Lie groups (classical solutions of the chiral model). J. Diff. Geom. 30, 1–50 (1989)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rui Pacheco.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pacheco, R. Bianchi–Bäcklund transforms and dressing actions, revisited. Geom Dedicata 146, 85–99 (2010). https://doi.org/10.1007/s10711-009-9427-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-009-9427-5

Keywords

Mathematics Subject Classification (2000)

Navigation